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Journal articles on the topic 'Monoid'

Author: Grafiati

Published: 4 June 2021

Last updated: 11 March 2023

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1

Ceccherini-Silberstein, Tullio, and Michel Coornaert. "On surjunctive monoids." International Journal of Algebra and Computation 25, no. 04 (May 21, 2015): 567–606. http://dx.doi.org/10.1142/s0218196715500113.

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A monoid M is called surjunctive if every injective cellular automata with finite alphabet over M is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all residually finite monoids, all finitely generated linear monoids, and all cancellative one-sided amenable monoids are surjunctive. We also prove that every limit of marked surjunctive monoids is itself surjunctive. On the other hand, we show that the bicyclic monoid and, more generally, all monoids containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive.

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2

Kim, Hwankoo, Myeong Og Kim, and Young Soo Park. "Some Characterizations of Krull Monoids." Algebra Colloquium 14, no. 03 (September 2007): 469–77. http://dx.doi.org/10.1142/s1005386707000429.

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In this paper, Kaplansky-type theorems are given to characterize GCD-monoids and valuation monoids. Also, (unique) r-factorable monoids are defined and it is shown that S is a Krull monoid if and only if S is a unique t-factorable (resp., w-factorable) monoid if and only if S is a t-factorable (resp., w-factorable) t-Prüfer monoid.

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3

Lee, Edmond. "Varieties generated by 2-testable monoids." Studia Scientiarum Mathematicarum Hungarica 49, no. 3 (September 1, 2012): 366–89. http://dx.doi.org/10.1556/sscmath.49.2012.3.1211.

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The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

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4

Polo, Harold. "Approximating length-based invariants in atomic Puiseux monoids." Algebra and Discrete Mathematics 33, no. 1 (2022): 128–39. http://dx.doi.org/10.12958/adm1760.

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A numerical monoid is a cofinite additive submonoid of the nonnegative integers, while a Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. Using that a Puiseux monoid is an increasing union of copies of numerical monoids, we prove that some of the factorization invariants of these two classes of monoids are related through a limiting process. This allows us to extend results from numerical to Puiseux monoids. We illustrate the versatility of this technique by recovering various known results about Puiseux monoids.

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Huang, W., and J. Li. "Affinely spanned quasi-stochastic algebraic monoids." International Journal of Algebra and Computation 27, no. 08 (December 2017): 1061–72. http://dx.doi.org/10.1142/s0218196717500497.

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A linear algebraic monoid over an algebraically closed field [Formula: see text] of characteristic zero is called (row) quasi-stochastic if each row of each matrix element is of sum one. Any linear algebraic monoid over [Formula: see text] can be embedded as an algebraic submonoid of the maximum affinely spanned quasi-stochastic monoid of some degree [Formula: see text]. The affinely spanned quasi-stochastic algebraic monoids form a basic class of quasi-stochastic algebraic monoids. An initial study of structure of affinely spanned quasi-stochastic algebraic monoids is conducted. Among other things, it is proved that the Zariski closure of a parabolic subgroup of the unit group of an affinely spanned quasi-stochastic algebraic monoid is affinely spanned.

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Cain, Alan J., and António Malheiro. "Deciding conjugacy in sylvester monoids and other homogeneous monoids." International Journal of Algebra and Computation 25, no. 05 (August 2015): 899–915. http://dx.doi.org/10.1142/s0218196715500241.

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We give a combinatorial characterization of conjugacy in the sylvester monoid, showing that conjugacy is decidable for this monoid. We then prove that conjugacy is undecidable in general for homogeneous monoids and even for multihomogeneous monoids.

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7

Schwab, Emil Daniel. "Gauge Inverse Monoids." Algebra Colloquium 27, no. 02 (May 7, 2020): 181–92. http://dx.doi.org/10.1142/s1005386720000152.

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The paper introduces a class of inverse (sub)monoids which contains Jones–Lawson’s gauge inverse (sub)monoid. The aim is to give examples and the basic properties of these monoids. Jones–Lawson’s gauge inverse monoid, as an inverse submonoid of the polycyclic monoid, is the prototype in our development line. The generalization leads also to Meakin–Sapir type results involving bijections between special congruences and special wide inverse submonoids.

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Cain, Alan J., António Malheiro, and Fábio M. Silva. "The monoids of the patience sorting algorithm." International Journal of Algebra and Computation 29, no. 01 (February 2019): 85–125. http://dx.doi.org/10.1142/s0218196718500649.

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The left patience sorting ([Formula: see text][Formula: see text]PS) monoid, also known in the literature as the Bell monoid, and the right patient sorting ([Formula: see text]PS) monoid are introduced by defining certain congruences on words. Such congruences are constructed using insertion algorithms based on the concept of decreasing subsequences. Presentations for these monoids are given. Each finite-rank [Formula: see text]PS monoid is shown to have polynomial growth and to satisfy a nontrivial identity (dependent on its rank), while the infinite rank [Formula: see text]PS monoid does not satisfy any nontrivial identity. Each [Formula: see text][Formula: see text]PS monoid of finite rank has exponential growth and does not satisfy any nontrivial identity. The complexity of the insertion algorithms is discussed. [Formula: see text]PS monoids of finite rank are shown to be automatic and to have recursive complete presentations. When the rank is [Formula: see text] or [Formula: see text], they are also biautomatic. [Formula: see text][Formula: see text]PS monoids of finite rank are shown to have finite complete presentations and to be biautomatic.

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9

Behrisch, Mike, and Edith Vargas-García. "Centralising Monoids with Low-Arity Witnesses on a Four-Element Set." Symmetry 13, no. 8 (August 11, 2021): 1471. http://dx.doi.org/10.3390/sym13081471.

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As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Mal’cev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal.

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10

GABBAY, MURDOCH J., and PETER H. KROPHOLLER. "Imaginary groups: lazy monoids and reversible computation." Mathematical Structures in Computer Science 23, no. 5 (May 15, 2013): 1002–31. http://dx.doi.org/10.1017/s0960129512000849.

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We use constructions in monoid and group theory to exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms and the category of partially ordered groups and group homomorphisms such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups.We introduce the new notion of a lazy homomorphism for a function f between partially ordered monoids such that f(m ○ m′) ≤ f(m) ○ f(m′).Every monoid can be endowed with the discrete partial ordering (m ≤ m′ if and only if m=m′), so our constructions provide a way of embedding monoids into groups. A simple counterexample (the two-element monoid with a non-trivial idempotent) and some calculations show that one can never hope for such an embedding to be a monoid homomorphism, so the price paid for injecting a monoid into a group is that we must weaken the notion of a homomorphism to this new notion of a lazy homomorphism.The computational significance of this is that a monoid is an abstract model of computation – or at least of ‘operations’ – and, similarly, a group models reversible computations/operations. With this reading, the adjunction with its injective unit gives a systematic high-level way of faithfully translating an irreversible system into a ‘lazy’ reversible one.Informally, but perhaps informatively, we can describe this work as follows: we give an abstract analysis of how we can sensibly add ‘undo’ (in the sense of ‘control-Z’).

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Stepanova, A. A. "S-acts over a Well-ordered Monoid with Modular Congruence Lattice." Bulletin of Irkutsk State University. Series Mathematics 35 (2021): 87–102. http://dx.doi.org/10.26516/1997-7670.2021.35.87.

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This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left $S$act over monoid $S$ is a set $A$ upon which $S$ acts unitarily on the left. In this paper, we consider $S$acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid $S$ is a linearly ordered set with a minimal element and with a binary operation $ \ max$, with respect to which $S$ is obviously a commutative monoid; a well-ordered monoid $S$ is a well-ordered set with a binary operation $ \ max$, with respect to which $S$ is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with M.S. Kazak, which describes $S$acts over linearly ordered monoids with a linearly ordered congruence lattice and $S$-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.

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12

GOULD, VICTORIA. "GRAPH EXPANSIONS OF RIGHT CANCELLATIVE MONOIDS." International Journal of Algebra and Computation 06, no. 06 (December 1996): 713–33. http://dx.doi.org/10.1142/s0218196796000404.

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The relations ℛ* and [Formula: see text] on a monoid M are natural generalizations of Green’s relations ℛ and [Formula: see text], which coincide with ℛ and [Formula: see text] if M is regular. A monoid M in which every ℛ*-class [Formula: see text] contains an idempotent is called left (right) abundant; if in addition the idempotents of M commute, that is, E(M) is a semilattice, then M is left (right) adequate. Regular monoids are obviously left (and right) abundant and inverse monoids are left (and right) adequate. Many of the well known results of regular and inverse semigroup theory have analogues for left abundant and left adequate monoids, or at least to special classes thereof. The aim of this paper is to develop a construction of left adequate monoids from the Cayley graph of a presentation of a right cancellative monoid, inspired by the construction of inverse monoids from group presentations, given by Margolis and Meakin in [10]. This technique yields in particular the free left ample (formerly left type A) monoid on a given set X.

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13

Gomez, Antonio Cano, and Magnus Steinby. "GENERALIZED CONTEXTS AND n-ARY SYNTACTIC SEMIGROUPS OF TREE LANGUAGES." Asian-European Journal of Mathematics 04, no. 01 (March 2011): 49–79. http://dx.doi.org/10.1142/s179355711100006x.

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A new type of syntactic monoid and semigroup of tree languages is introduced. For each n ≥ 1, we define for any tree language T its n-ary syntactic monoid Mn(T) and its n-ary syntactic semigroup Sn(T) as quotients of the monoid or semigroup, respectively, formed by certain new generalized contexts. For n = 1 these contexts are just the ordinary contexts (or 'special trees') and M1(T) is the syntactic monoid introduced by W. Thomas (1982,1984). Several properties of these monoids and semigroups are proved. For example, it is shown that Mn(T) and Sn(T) are isomorphic to certain monoids and semigroups associated with the minimal tree recognizer of T. Using these syntactic monoids or semigroups, we can associate with any variety of finite monoids or semigroups, respectively, a variety of tree languages. Although there are varieties of tree languages that cannot be obtained this way, we prove that the definite tree languages can be characterized by the syntactic semigroups S2(T), which is not possible using the classical syntactic monoids or semigroups.

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14

GODELLE, EDDY. "PRESENTATION FOR RENNER MONOIDS." Bulletin of the Australian Mathematical Society 83, no. 1 (June 22, 2010): 30–45. http://dx.doi.org/10.1017/s0004972710000365.

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AbstractWe extend the result obtained in E. Godelle [‘The braid rook monoid’, Internat. J. Algebra Comput.18 (2008), 779–802] to every Renner monoid: we provide a monoid presentation for Renner monoids, and we introduce a length function which extends the Coxeter length function and which behaves nicely.

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15

Zhang, Louxin. "Applying rewriting methods to special monoids." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 3 (November 1992): 495–505. http://dx.doi.org/10.1017/s0305004100071176.

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A special monoid is a monoid presented by generators and defining relations of the form w = e, where w is a non-empty word on generators and e is the empty word. Groups are special monoids. But there exist special monoids that are not groups. Special monoids have been extensively studied by Adjan[1] and Makanin[7] (see also [2]).

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16

BRANCO, MÁRIO J. J., GRACINDA M. S. GOMES, and VICTORIA GOULD. "LEFT ADEQUATE AND LEFT EHRESMANN MONOIDS." International Journal of Algebra and Computation 21, no. 07 (November 2011): 1259–84. http://dx.doi.org/10.1142/s0218196711006935.

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This is the first of two articles studying the structure of left adequate and, more generally, of left Ehresmann monoids. Motivated by a careful analysis of normal forms, we introduce here a concept of proper for a left adequate monoid M. In fact, our notion is that of T-proper, where T is a submonoid of M. We show that any left adequate monoid M has an X*-proper cover for some set X, that is, there is a left adequate monoid [Formula: see text] that is X*-proper, and an idempotent separating surjective morphism [Formula: see text] of the appropriate type. Given this result, we may deduce that the free left adequate monoid on any set X is X*-proper. In a subsequent paper, we show how to construct T-proper left adequate monoids from any monoid T acting via order-preserving maps on a semilattice with identity, and prove that the free left adequate monoid is of this form. An alternative description of the free left adequate monoid will appear in a paper of Kambites. We show how to obtain the labeled trees appearing in his result from our structure theorem. Our results apply to the wider class of left Ehresmann monoids, and we give them in full generality. We also indicate how to obtain some of the analogous results in the two-sided case. This paper and its sequel, and the two of Kambites on free (left) adequate semigroups, demonstrate the rich but accessible structure of (left) adequate semigroups and monoids, introduced with startling insight by Fountain some 30 years ago.

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17

V.K, Sreeja. "Construction of Inverse Unit Regular Monoids from a Semilattice and a Group." International Journal of Engineering & Technology 7, no. 4.36 (December 9, 2018): 950. http://dx.doi.org/10.14419/ijet.v7i4.36.24927.

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This paper is a continuation of a previous paper [6] in which the structure of certain unit regular semigroups called R-strongly unit regular monoids has been studied. A monoid S is said to be unit regular if for each element s Î S there exists an element u in the group of units G of S such that s = sus. Hence where su is an idempotent and is a unit. A unit regular monoid S is said to be a unit regular inverse monoid if the set of idempotents of S form a semilattice. Since inverse monoids are R unipotent, every element of a unit regular inverse monoid can be written as s = eu where the idempotent part e is unique and u is a unit. Here we give a detailed study of inverse unit regular monoids and the results are mainly based on [10]. The relations between the semilattice of idempotents and the group of units in unit regular inverse monoids are better identified in this case. .

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18

GODELLE, EDDY. "THE BRAID ROOK MONOID." International Journal of Algebra and Computation 18, no. 04 (June 2008): 779–802. http://dx.doi.org/10.1142/s0218196708004603.

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In linear algebraic monoid theory, the Renner monoids play the role of the Weyl groups in linear algebraic group theory. It is well known that Weyl groups are Coxeter groups, and that we can associate a Hecke algebra and an Artin–Tits group to each Coxeter group. The question of the existence of a Hecke algebra associated with each Renner monoid has been positively answered. In this paper we discuss the question of the existence of an equivalent of the Artin–Tits groups in the framework of Renner monoids. We consider the seminal case of the rook monoid and introduce a new length function.

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SCHMID, WOLFGANG A. "HIGHER-ORDER CLASS GROUPS AND BLOCK MONOIDS OF KRULL MONOIDS WITH TORSION CLASS GROUP." Journal of Algebra and Its Applications 09, no. 03 (June 2010): 433–64. http://dx.doi.org/10.1142/s0219498810004002.

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Extensions of the notion of a class group and a block monoid associated to a Krull monoid with torsion class group are introduced and investigated. Instead of assigning to a Krull monoid only one abelian group (the class group) and one monoid of zero-sum sequences (the block monoid), we assign to it a recursively defined family of abelian groups, the first being the class group, and do alike for the block monoid. These investigations are motivated by the aim of gaining a more detailed understanding of the arithmetic of Krull monoids, including Dedekind and Krull domains, both from a technical and conceptual point of view. To illustrate our method, some first arithmetical applications are presented.

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20

Bardakov, Valeriy G., Slavik Jablan, and Hang Wang. "Monoid and group of pseudo braids." Journal of Knot Theory and Its Ramifications 25, no. 09 (August 2016): 1641002. http://dx.doi.org/10.1142/s0218216516410029.

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21

LAWSON, MARK V. "A CORRESPONDENCE BETWEEN BALANCED VARIETIES AND INVERSE MONOIDS." International Journal of Algebra and Computation 16, no. 05 (October 2006): 887–924. http://dx.doi.org/10.1142/s0218196706003165.

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There is a well-known correspondence between varieties of algebras and fully invariant congruences on the appropriate term algebra. A special class of varieties are those which are balanced, meaning they can be described by equations in which the same variables appear on each side. In this paper, we prove that the above correspondence, restricted to balanced varieties, leads to a correspondence between balanced varieties and inverse monoids. In the case of unary algebras, we recover the theorem of Meakin and Sapir that establishes a bijection between congruences on the free monoid with n generators and wide, positively self-conjugate inverse submonoids of the polycyclic monoid on n generators. In the case of varieties generated by linear equations, meaning those equations where each variable occurs exactly once on each side, we can replace the clause monoid above by the linear clause monoid. In the case of algebras with a single operation of arity n, we prove that the linear clause monoid is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated essential right ideals of the free monoid on n letters, a monoid previously studied by Birget in the course of work on the Thompson group V and its analogues. We show that Dehornoy's geometry monoid of a balanced variety is a special kind of inverse submonoid of ours. Finally, we construct groups from the inverse monoids associated with a balanced variety and examine some conditions under which they still reflect the structure of the underlying variety. Both free groups and Thompson's groups Vn,1 arise in this way.

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22

FitzGerald, D. G. "A presentation for the monoid of uniform block permutations." Bulletin of the Australian Mathematical Society 68, no. 2 (October 2003): 317–24. http://dx.doi.org/10.1017/s0004972700037692.

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The monoid n of uniform block permutations is the factorisable inverse monoid which arises from the natural action of the symmetric group on the join semilattice of equivalences on an n-set; it has been described in the literature as the factorisable part of the dual symmetric inverse monoid. The present paper gives and proves correct a monoid presentation forn. The methods involved make use of a general criterion for a monoid generated by a group and an idempotent to be inverse, the structure of factorisable inverse monoids, and presentations of the symmetric group and the join semilattice of equivalences on an n-set.

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23

ROSALES, J. C., P. A. GARCÍA-SÁNCHEZ, and J. M. URBANO-BLANCO. "ON PRESENTATIONS OF COMMUTATIVE MONOIDS." International Journal of Algebra and Computation 09, no. 05 (October 1999): 539–53. http://dx.doi.org/10.1142/s0218196799000333.

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In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids. As a consequence, we give a method to obtain a presentation for any commutative monoid.

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GILBERT, N. D., and R. NOONAN HEALE. "THE IDEMPOTENT PROBLEM FOR AN INVERSE MONOID." International Journal of Algebra and Computation 21, no. 07 (November 2011): 1179–94. http://dx.doi.org/10.1142/s0218196711006893.

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We generalize the word problem for groups, the formal language of all words in the generators that represent the identity, to inverse monoids. In particular, we introduce the idempotent problem, the formal language of all words representing idempotents, and investigate how the properties of an inverse monoid are related to the formal language properties of its idempotent problem. We show that if an inverse monoid is either E-unitary or has a finite set of idempotents, then its idempotent problem is regular if and only if the inverse monoid is finite. We also give examples of inverse monoids with context-free idempotent problems, including all Bruck–Reilly extensions of finite groups.

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Köcher, Chris, Dietrich Kuske, and Olena Prianychnykova. "The inclusion structure of partially lossy queue monoids and their trace submonoids." RAIRO - Theoretical Informatics and Applications 52, no. 1 (January 2018): 55–86. http://dx.doi.org/10.1051/ita/2018003.

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We model the behavior of a lossy fifo-queue as a monoid of transformations that are induced by sequences of writing and reading. To have a common model for reliable and lossy queues, we split the alphabet of the queue into two parts: the forgettable letters and the letters that are transmitted reliably. We describe this monoid by means of a confluent and terminating semi-Thue system and then study some of the monoid’s algebraic properties. In particular, we characterize completely when one such monoid can be embedded into another as well as which trace monoids occur as submonoids. Surprisingly, these are precisely those trace monoids that embed into the direct product of two free monoids – which gives a partial answer to a question raised by Diekert et al. at STACS 1995.

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FITZGERALD, D. G., and KWOK WAI LAU. "ON THE PARTITION MONOID AND SOME RELATED SEMIGROUPS." Bulletin of the Australian Mathematical Society 83, no. 2 (December 6, 2010): 273–88. http://dx.doi.org/10.1017/s0004972710001851.

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AbstractThe partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

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EAST, JAMES. "EMBEDDINGS IN COSET MONOIDS." Journal of the Australian Mathematical Society 85, no. 1 (August 2008): 75–80. http://dx.doi.org/10.1017/s1446788708000153.

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AbstractA submonoid S of a monoid M is said to be cofull if it contains the group of units of M. We extract from the work of Easdown, East and FitzGerald (2002) a sufficient condition for a monoid to embed as a cofull submonoid of the coset monoid of its group of units, and show further that this condition is necessary. This yields a simple description of the class of finite monoids which embed in the coset monoids of their group of units. We apply our results to give a simple proof of the result of McAlister [D. B. McAlister, ‘Embedding inverse semigroups in coset semigroups’, Semigroup Forum20 (1980), 255–267] which states that the symmetric inverse semigroup on a finite set X does not embed in the coset monoid of the symmetric group on X. We also explore examples, which are necessarily infinite, of embeddings whose images are not cofull.

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Kudryavtseva, Ganna. "Two-sided expansions of monoids." International Journal of Algebra and Computation 29, no. 08 (October 24, 2019): 1467–98. http://dx.doi.org/10.1142/s0218196719500590.

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We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget–Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of relatively free inverse monoids. For a monoid [Formula: see text] and a class of partial actions of [Formula: see text], determined by a set, [Formula: see text], of identities, we define [Formula: see text] to be the universal [Formula: see text]-generated two-sided restriction monoid with respect to partial actions of [Formula: see text] determined by [Formula: see text]. This is an [Formula: see text]-restriction monoid which (for a certain [Formula: see text]) generalizes the Birget–Rhodes prefix expansion [Formula: see text] of a group [Formula: see text]. Our main result provides a coordinatization of [Formula: see text] via a partial action product of the idempotent semilattice [Formula: see text] of a similarly defined inverse monoid, partially acted upon by [Formula: see text]. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result. We show that some properties of [Formula: see text] agree well with suitable properties of [Formula: see text], such as being cancellative or embeddable into a group. We observe that if [Formula: see text] is an inverse monoid, then [Formula: see text], the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson–Margolis–Steinberg generalized prefix expansion [Formula: see text]. This gives a presentation of [Formula: see text] and leads to a model for [Formula: see text] in terms of the known model for [Formula: see text].

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KAARLI, KALLE, and LÁSZLÓ MÁRKI. "A CHARACTERIZATION OF THE INVERSE MONOID OF BI-CONGRUENCES OF CERTAIN ALGEBRAS." International Journal of Algebra and Computation 19, no. 06 (September 2009): 791–808. http://dx.doi.org/10.1142/s021819670900538x.

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This paper provides an abstract characterization of the inverse monoids that appear as monoids of bi-congruences of finite minimal algebras generating arithmetical varieties. As a tool, a matrix construction is introduced which might be of independent interest in inverse semigroup theory. Using this construction as well as Ramsey's theorem, we embed a certain kind of inverse monoid into a factorizable monoid of the same kind. As noticed by M. Lawson, this embedding entails that the embedded finite monoids have finite F-unitary cover.

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Dubey, M. K., and S. P. Tiwari. "The Relationship Among Fuzzy Languages, Upper Sets and Fuzzy Ordered Monoids." New Mathematics and Natural Computation 15, no. 02 (June 20, 2019): 361–72. http://dx.doi.org/10.1142/s1793005719500200.

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The concept of recognizability of a fuzzy language by crisp deterministic fuzzy automaton and monoid is well known. The purpose of the present work is to study the recognizability of fuzzy languages by fuzzy ordered monoids. We show that the upper sets of a given fuzzy ordered monoid play a nice role in such studies. Also, we introduce the syntactic fuzzy ordered monoid of an upper set which recognizes this upper set.

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31

COJUHARI, E. P., and B. J. GARDNER. "GENERALIZED HIGHER DERIVATIONS." Bulletin of the Australian Mathematical Society 86, no. 2 (January 6, 2012): 266–81. http://dx.doi.org/10.1017/s000497271100308x.

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AbstractA type of generalized higher derivation consisting of a collection of self-mappings of a ring associated with a monoid, and here called a D-structure, is studied. Such structures were previously used to define various kinds of ‘skew’ or ‘twisted’ monoid rings. We show how certain gradings by monoids define D-structures. The monoid ring defined by such a structure corresponding to a group-grading is the variant of the group ring introduced by Năstăsescu, while in the case of a cyclic group of order two, the form of the D-structure itself yields some gradability criteria of Bakhturin and Parmenter. A partial description is obtained of the D-structures associated with infinite cyclic monoids.

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KOUBEK, VÁCLAV, VOJTĚCH RÖDL, and BENJAMIN SHEMMER. "REPRESENTING SUBDIRECT PRODUCT MONOIDS BY GRAPHS." International Journal of Algebra and Computation 19, no. 05 (August 2009): 705–21. http://dx.doi.org/10.1142/s0218196709005275.

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Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. In a previous paper, we give a construction G(M) for a graph with prescribed endomorphism monoid M known as a [Formula: see text]-graph. Using this construction, we derived bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. In this paper, we generalize the [Formula: see text]-graph construction and derive several new bounds for monoid classes not handled by our first paper. Among these are the so called "strong semilattices of C-semigroups" where C is one of the following: Groups, Abelian Groups, Rectangular Groups, and completely simple semigroups.

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KUDRYAVTSEVA, GANNA, and VOLODYMYR MAZORCHUK. "PARTIALIZATION OF CATEGORIES AND INVERSE BRAID-PERMUTATION MONOIDS." International Journal of Algebra and Computation 18, no. 06 (September 2008): 989–1017. http://dx.doi.org/10.1142/s0218196708004731.

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We show how one can use the partialization functor to obtain several recently defined inverse monoids, and use this functor to define new objects, which we call the inverse braid-permutation monoids. A presentation for this monoid is obtained. Finally, we study some abstract properties of the partialization functor and its iterations. This leads to a categorification of a monoid of all order-preserving maps, and series of orthodox generalizations of the symmetric inverse semigroup.

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DEIS, TIMOTHY, JOHN MEAKIN, and G. SÉNIZERGUES. "EQUATIONS IN FREE INVERSE MONOIDS." International Journal of Algebra and Computation 17, no. 04 (June 2007): 761–95. http://dx.doi.org/10.1142/s0218196707003755.

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It is known that the problem of determining consistency of a finite system of equations in a free group or a free monoid is decidable, but the corresponding problem for systems of equations in a free inverse monoid of rank at least two is undecidable. Any solution to a system of equations in a free inverse monoid induces a solution to the corresponding system of equations in the associated free group in an obvious way, but solutions to systems of equations in free groups do not necessarily lift to solutions in free inverse monoids. In this paper, we show that the problem of determining whether a solution to a finite system of equations in a free group can be extended to a solution of the corresponding system in the associated free inverse monoid is decidable. We are able to use this to solve the consistency problem for certain classes of single-variable equations in free inverse monoids.

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35

Aguiar, Marcelo, and Swapneel Mahajan. "On the Hadamard Product of Hopf Monoids." Canadian Journal of Mathematics 66, no. 3 (June 1, 2014): 481–504. http://dx.doi.org/10.4153/cjm-2013-005-x.

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AbstractCombinatorial structures that compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free.The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco that applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species that is intertwined by the free monoid functor with theHadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.

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ZHANG, LOUXIN. "ON THE CONJUGACY PROBLEM FOR ONE-RELATOR MONOIDS WITH ELEMENTS OF FINITE ORDER." International Journal of Algebra and Computation 02, no. 02 (June 1992): 209–20. http://dx.doi.org/10.1142/s021819679200013x.

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A one-relator monoid has nontrivial elements of finite order if and only if its presentation has the form (A; (PQ)mP=(PQ)nP), where PQ is a primitive word and m>n≥0. The (left-)conjugacy problem for such a monoid is shown to be reducible to the same problem for its left monoid. In particular, the (left-)conjugacy problem is decidable for the monoids M(A;(PQ)mP=(PQ)nP), where m+n≥2.

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Ateş, Firat, Eylem G. Karpuz, A. Dilek Güngör, and A. Sinan Çevik. "A NEW EXAMPLE FOR MINIMALITY OF MONOIDS." Asian-European Journal of Mathematics 03, no. 04 (December 2010): 531–44. http://dx.doi.org/10.1142/s1793557110000416.

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By considering the split extension of a free abelian monoid having finite rank by a finite monogenic monoid, the main purposes of this paper are to present examples of efficient monoids and, also, minimal but inefficient monoids. Although results presented in this paper seem as in the branch of pure mathematics, they are actually related to applications of Combinatorial and Geometric Group-Semigroup Theory, especially computer science, network systems, cryptography and physics etc., which will not be handled here.

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Fountain, John, and Gracinda M. S. Gomes. "PROPER COVERS OF AMPLE MONOIDS." Proceedings of the Edinburgh Mathematical Society 49, no. 2 (May 30, 2006): 277–89. http://dx.doi.org/10.1017/s0013091504000070.

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AbstractProper ample monoids are described by means of a certain category acted upon on both sides by a cancellative monoid. Making use of this characterization, we show that every ample monoid $S$ has a proper ample cover, which can be taken to be finite whenever $S$ is finite.

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LOHREY, MARKUS. "DECIDABILITY AND COMPLEXITY IN AUTOMATIC MONOIDS." International Journal of Foundations of Computer Science 16, no. 04 (August 2005): 707–22. http://dx.doi.org/10.1142/s0129054105003248.

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Several complexity and decidability results for automatic monoids are shown: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the first-order theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Moreover, it is shown that for every hyperbolic group the word problem belongs to LOGCFL, which improves a result of Cai [8].

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40

Crabb, M. J., and W. D. Munn. "On the l1-algebra of certain monoids." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 5 (1998): 1023–31. http://dx.doi.org/10.1017/s0308210500030043.

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The monoids considered are the free monoid Mx and the free monoid-with-involution MIx on a nonempty set X. In each case, relative to a simply-defined involution, an explicit construction is given for a separating family of continuous star matrix representations of the l1-algebra of the monoid and it is shown that this algebra admits a faithful trace. The results are based on earlier work by M. J. Crabb et al. concerning the complex semigroup algebras of Mx and MIx.

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MARGOLIS, STUART W., and JOHN C. MEAKIN. "FREE INVERSE MONOIDS AND GRAPH IMMERSIONS." International Journal of Algebra and Computation 03, no. 01 (March 1993): 79–99. http://dx.doi.org/10.1142/s021819679300007x.

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The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

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42

GOULD, VICTORIA, and MIKLÓS HARTMANN. "Coherency, free inverse monoids and related free algebras." Mathematical Proceedings of the Cambridge Philosophical Society 163, no. 1 (September 9, 2016): 23–45. http://dx.doi.org/10.1017/s0305004116000505.

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AbstractA monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is the non-additive notion corresponding to that for a ring R stating that every finitely generated submodule of every finitely presented right R-module is finitely presented. For monoids (and rings) right coherency is an important finitary property which determines, amongst other things, the existence of a model companion of the class of right S-acts (right R-modules) and hence that the class of existentially closed right S-acts (right R-modules) is axiomatisable.Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruškuc, have shown that (free) groups, free commutative monoids and free monoids have the same properties. It is then natural to ask whether other free algebras in varieties of monoids, possibly with an augmented signature, are right coherent. We demonstrate that free inverse monoids are not.Munn described the free inverse monoid FIM(Ω) on Ω as consisting of birooted finite connected subgraphs of the Cayley graph of the free group on Ω. Sitting within FIM(Ω) we have free algebras in other varieties and quasi-varieties, in particular the free left ample monoid FLA(Ω) and the free ample monoid FAM(Ω). The former is the free algebra in the variety of unary monoids corresponding to partial maps with distinguished domain; the latter is the two-sided dual. For example, FLA(Ω) is obtained from FIM(Ω) by considering only subgraphs with vertices labelled by elements of the free monoid on Ω.The main objective of the paper is to show that FLA(Ω) is right coherent. Furthermore, by making use of the same techniques we show that FIM(Ω), FLA(Ω) and FAM(Ω) satisfy (R), (r), (L) and (l), related conditions arising from the axiomatisability of certain classes of right S-acts and of left S-acts.

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43

Gould, Victoria. "Coherent Monoids." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 53, no. 2 (October 1992): 166–82. http://dx.doi.org/10.1017/s1446788700035771.

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AbstractThis paper is concerned with a new notion of coherency for monoids. A monoid S is right coherent if the first order theory of right S-sets is coherent; this is equivalent to the property that every finitely generated S-subset of every finitely presented right S-set is finitely presented. If every finitely generated right ideal of S is finitely presented we say that S is weakly right coherent. As for the corresponding situation for modules over a ring, we show that our notion of coherency is related to products of flat left S-sets, although there are some marked differences in behaviour from the case for rings. Further, we relate our work to ultraproducts of flat left S-sets and so to the question of axiomatisability of certain classes of left S-sets.We show that a monoid S is weakly right coherent if and only if the right annihilator congruence of every element is finitely generated and the intersection of any two finitely generated right ideals is finitely generated. A similar result describes right coherent monoids. We use these descriptions to recognise several classes of (weakly) right coherent monoids. In particular we show that any free monoid is weakly right (and left) coherent and any free commutative monoid is right (and left) coherent.

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44

Alonso, Juan M., and Susan M. Hermiller. "Homological Finite Derivation Type." International Journal of Algebra and Computation 13, no. 03 (June 2003): 341–59. http://dx.doi.org/10.1142/s0218196703001407.

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In 1987, Squier defined the notion of finite derivation type for a finitely presented monoid. To do this, he associated a 2-complex to the presentation. The monoid then has finite derivation type if, modulo the action of the free monoid ring, the 1-dimensional homotopy of this complex is finitely generated. Cremanns and Otto showed that finite derivation type implies the homological finiteness condition left FP3, and when the monoid is a group, these two properties are equivalent. In this paper we define a new version of finite derivation type, based on homological information, together with an extension of this finite derivation type to higher dimensions, and show connections to homological type FPnfor both monoids and groups.

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Averbukh, Boris G. "A criterion of the existence of an embedding of a monothetic monoid into a topological group." Topological Algebra and its Applications 7, no. 1 (May 16, 2019): 1–12. http://dx.doi.org/10.1515/taa-2019-0001.

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AbstractUsing properties of unitary Cauchy filters on monothetic monoids, we prove a criterion of the existence of an embedding of such a monoid into a topological group. The proof of the sufficiency is constructive: under the corresponding assumptions, we are building a dense embedding of a given monothetic monoid into a monothetic group.

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46

Corson, Jon M., and Lance L. Ross. "Automata with Counters that Recognize Word Problems of Free Products." International Journal of Foundations of Computer Science 26, no. 01 (January 2015): 79–98. http://dx.doi.org/10.1142/s0129054115500045.

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An M-automaton is a finite automaton with a blind counter that mimics a monoid M. The finitely generated groups whose word problems (when viewed as formal languages) are accepted by M-automata play a central role in understanding the family 𝔏(M) of all languages accepted by M-automata. If G1 and G2 are finitely generated groups whose word problems are languages in 𝔏(M), in general, the word problem of the free product G1 * G2 is not necessarily in 𝔏(M). However, we show that if M is enlarged to the free product M*P2, where P2 is the polycyclic monoid of rank two, then this closure property holds. In fact, we show more generally that the special word problem of M1 * M2 lies in 𝔏(M * P2) whenever M1 and M2 are finitely generated monoids with special word problems in 𝔏(M * P2). We also observe that there is a monoid without zero, denoted by CF2, that can be used in place of P2 for this purpose. The monoid CF2 is the rank two case of what we call a monoid with right invertible basis and its Rees quotient by its maximal ideal is P2. The fundamental theory of monoids with right invertible bases is completely analogous to that of free groups, and thus they are very convenient to use. We also investigate the questions of whether there is a group that can be used instead of the monoid P2 in the above result and under what circumstances P1 (or the bicyclic monoid) is enough to do the job of P2.

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47

Lee, Edmond W. H. "Varieties of involution monoids with extreme properties." Quarterly Journal of Mathematics 70, no. 4 (June 22, 2019): 1157–80. http://dx.doi.org/10.1093/qmath/haz003.

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Abstract A variety that contains continuum many subvarieties is said to be huge. A sufficient condition is established under which an involution monoid generates a variety that is huge by virtue of its lattice of subvarieties order-embedding the power set lattice of the positive integers. Based on this result, several examples of finite involution monoids with extreme varietal properties are exhibited. These examples—all first of their kinds—include the following: finite involution monoids that generate huge varieties but whose reduct monoids generate Cross varieties; two finite involution monoids sharing a common reduct monoid such that one generates a huge, non-finitely based variety while the other generates a Cross variety; and two finite involution monoids that generate Cross varieties, the join of which is huge.

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LI, JIAN RONG, and YAN FENG LUO. "EQUATIONAL PROPERTY OF CERTAIN TRANSFORMATION MONOIDS." International Journal of Algebra and Computation 20, no. 06 (September 2010): 833–45. http://dx.doi.org/10.1142/s0218196710005947.

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The monoid of extensive transformations of a chain of order four is shown to be finitely based and a finite basis for this monoid is given. This completes the description of the equational property of the monoids of all full extensive transformations, partial extensive transformations, and partial order-preserving extensive transformations over any finite chain.

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49

Heath, Philip R. "Lifting amalgams and other colimits of monoids." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 1 (July 1990): 21–29. http://dx.doi.org/10.1017/s0305004100068912.

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Let U = [{Mi: i ∈ I}; U; {øi: i ∈ I}] be a monoid amalgam of inclusions, i.e. U and each Mi is a monoid, and øi: U → Mi are inclusions. Let be the monoid free product of the amalgam U (see for example [10] for these concepts), and let β: B → H be a homomorphism of monoids. The type of question we seek to answer in this paper is under what conditions (on β, B and H) can we deduce that B is isomorphic to the free product of the amalgam

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KAMBITES, MARK. "ON UNIFORM DECISION PROBLEMS AND ABSTRACT PROPERTIES OF SMALL OVERLAP MONOIDS." International Journal of Algebra and Computation 21, no. 01n02 (February 2011): 217–33. http://dx.doi.org/10.1142/s0218196711006145.

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We study the way in which the abstract structure of a small overlap monoid is reflected in, and may be algorithmically deduced from, a small overlap presentation. We show that every C(2) monoid admits an essentially canonical C(2) presentation; by counting canonical presentations we obtain asymptotic estimates for the number of non-isomorphic monoids admitting a-generator, k-relation presentations of a given length. We demonstrate an algorithm to transform an arbitrary presentation for a C(m) monoid (m at least 2) into a canonical C(m) presentation, and a solution to the isomorphism problem for C(2) presentations. We also find a simple combinatorial condition on a C(4) presentation which is necessary and sufficient for the monoid presented to be left cancellative. We apply this to obtain algorithms to decide if a given C(4) monoid is left cancellative, right cancellative or cancellative, and to show that cancellativity properties are asymptotically visible in the sense of generic-case complexity.

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