Relevant bibliographies by topics / Biordered set

Academic literature on the topic 'Biordered set'

Author: Grafiati
Published: 4 June 2021
Last updated: 21 February 2023

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Journal articles on the topic "Biordered set"

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Akhila, R., and P. G. Romeo. "On the lattice of biorder ideals of regular rings." Asian-European Journal of Mathematics 13, no. 06 (April 4, 2019): 2050103. http://dx.doi.org/10.1142/s179355712050103x.

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The study of biordered set plays a significant role in describing the structure of a regular semigroup and since the definition of regularity involves only the multiplication in the ring, it is natural that the study of semigroups plays a significant role in the study of regular rings. Here, we extend the biordered set approach to study the structure of the regular semigroup [Formula: see text] of a regular ring [Formula: see text] by studying the idempotents [Formula: see text] of the regular ring and show that the principal biorder ideals of the regular ring [Formula: see text] form a complemented modular lattice and certain properties of this lattice are studied.
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Dandan, Yang, and Victoria Gould. "Free idempotent generated semigroups over bands and biordered sets with trivial products." International Journal of Algebra and Computation 26, no. 03 (May 2016): 473–507. http://dx.doi.org/10.1142/s021819671650020x.

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For any biordered set of idempotents [Formula: see text] there is an initial object [Formula: see text], the free idempotent generated semigroup over[Formula: see text], in the category of semigroups generated by a set of idempotents biorder-isomorphic to [Formula: see text]. Recent research on [Formula: see text] has focused on the behavior of the maximal subgroups. Inspired by an example of Brittenham, Margolis and Meakin, several proofs have been offered that any group occurs as a maximal subgroup of some [Formula: see text], the latest being that of Dolinka and Ruškuc, who show that [Formula: see text] can be taken to be a band. From a result of Easdown, Sapir and Volkov, periodic elements of any [Formula: see text] lie in subgroups. However, little else is known of the “global” properties of [Formula: see text], other than that it need not be regular, even where [Formula: see text] is a semilattice. The aim of this paper is to deepen our understanding of the overall structure of [Formula: see text] in the case where [Formula: see text] is a biordered set with trivial products (for example, the biordered set of a poset) or where [Formula: see text] is the biordered set of a band [Formula: see text]. Since its introduction by Fountain in the late 1970s, the study of abundant and related semigroups has given rise to a deep and fruitful research area. The class of abundant semigroups extends that of regular semigroups in a natural way and itself is contained in the class of weakly abundant semigroups. Our main results show that (1) if [Formula: see text] is a biordered set with trivial products then [Formula: see text] is abundant and (if [Formula: see text] is finite) has solvable word problem, and (2) for any band [Formula: see text], the semigroup [Formula: see text] is weakly abundant and moreover satisfies a natural condition called the congruence condition. Further, [Formula: see text] is abundant for a normal band [Formula: see text] for which [Formula: see text] satisfies a given technical condition, and we give examples of such [Formula: see text]. On the other hand, we give an example of a normal band [Formula: see text] such that [Formula: see text] is not abundant.
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Rajan, A. R., and V. K. Sreeja. "CONSTRUCTION OF A R-STRONGLY UNIT REGULAR MONOID FROM A REGULAR BIORDERED SET AND A GROUP." Asian-European Journal of Mathematics 04, no. 04 (December 2011): 653–70. http://dx.doi.org/10.1142/s1793557111000526.

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In this paper we give a detailed study of R-strongly unit regular monoids. The relations between the biordered set of idempotents and the group of units in unit regular semigroups are better identified here. Conversely, starting from a regular biordered set E and a group G we construct a R-strongly unit regular semigroup S for which the set of idempotents E(S) is isomorphic to E as a biordered set and the group of units G(S) is isomorphic to G. The conditions to be satisfied by G and E are also listed.
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Yu, Bing Jun, and Mang Xu. "A Biordered Set Representation of Regular Semigroups." Acta Mathematica Sinica, English Series 21, no. 2 (February 18, 2005): 289–302. http://dx.doi.org/10.1007/s10114-004-0490-4.

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EASDOWN, D., M. V. SAPIR, and M. V. VOLKOV. "PERIODIC ELEMENTS OF THE FREE IDEMPOTENT GENERATED SEMIGROUP ON A BIORDERED SET." International Journal of Algebra and Computation 20, no. 02 (March 2010): 189–94. http://dx.doi.org/10.1142/s0218196710005583.

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Tamilarasi, A. "Idempotent-separating extensions of regular semigroups." International Journal of Mathematics and Mathematical Sciences 2005, no. 18 (2005): 2945–75. http://dx.doi.org/10.1155/ijmms.2005.2945.

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For a regular biordered setE, the notion ofE-diagram and the associated regular semigroup was introduced in our previous paper (1995). Given a regular biordered setE, anE-diagram in a categoryCis a collection of objects, indexed by the elements ofEand morphisms ofCsatisfying certain compatibility conditions. With such anE-diagramAwe associate a regular semigroupRegE(A)havingEas its biordered set of idempotents. This regular semigroup is analogous to automorphism group of a group. This paper provides an application ofRegE(A)to the idempotent-separating extensions of regular semigroups. We introduced the concept of crossed pair and used it to describe all extensions of a regular semigroup S by a groupE-diagramA. In this paper, the necessary and sufficient condition for the existence of an extension ofSbyAis provided. Also we study cohomology and obstruction theories and find a relationship with extension theory for regular semigroups.
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7

Ren, Xueming, Yanhui Wang, and K. P. Shum. "U-Concordant Semigroups." Algebra Colloquium 25, no. 02 (May 22, 2018): 295–318. http://dx.doi.org/10.1142/s1005386718000214.

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We introduce the relations [Formula: see text] and [Formula: see text] with respect to a subset U of idempotents. Based on [Formula: see text] and [Formula: see text], we define a new class of semigroups which we name U-concordant semigroups. Our purpose is to describe U-concordant semigroups by generalized categories over a regular biordered set. We show that the category of U-concordant semigroups and admissible morphisms is isomorphic to the category of RBS generalized categories and pseudo functors. Our approach is inspired from Armstrong’s work on the connection between regular biordered sets and concordant semigroups. The significant difference in strategy is by using RBS generalized categories equipped with pre-orders, we have no need to discuss the quotient of a category factored by a congruence.
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Gigoń, Roman S. "Completely simple congruences on E-inversive semigroups." Journal of Algebra and Its Applications 15, no. 06 (March 30, 2016): 1650052. http://dx.doi.org/10.1142/s0219498816500523.

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We study completely simple congruences on an arbitrary [Formula: see text]-inversive semigroup [Formula: see text]. In particular, we show that every such congruence [Formula: see text] on [Formula: see text] is uniquely determined by its kernel and trace, and that the trace of [Formula: see text] is a congruence on the biordered set [Formula: see text]. Moreover, we investigate the complete lattice of all completely simple congruences on [Formula: see text] and show that the trace relation is a complete congruence on this lattice. We also construct a family of completely simple congruences on [Formula: see text].
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9

McElwee, Brett. "SUBGROUPS OF THE FREE SEMIGROUP ON A BIORDERED SET IN WHICH PRINCIPAL IDEALS ARE SINGLETONS." Communications in Algebra 30, no. 11 (December 31, 2002): 5513–19. http://dx.doi.org/10.1081/agb-120015667.

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Easdown, David. "Biorder-preserving coextensions of fundamental semigroups." Proceedings of the Edinburgh Mathematical Society 31, no. 3 (October 1988): 463–67. http://dx.doi.org/10.1017/s0013091500037652.

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In any extension theory for semigroups one must determine the basic building blocks and then discover how they fit together to create more complicated semigroups. For example, in group theory the basic building blocks are simple groups. In semigroup theory however there are several natural choices. One that has received considerable attention, particularly since the seminal work on inverse semigroups by Munn ([14, 15]), is the notion of a fundamental semigroup. A semigroup is called fundamental if it cannot be [shrunk] homomorphically without collapsing some of its idempotents (see below for a precise definition).
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Dissertations / Theses on the topic "Biordered set"

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Roberts, Brad. "On bosets and fundamental semigroups." Thesis, The University of Sydney, 2007. http://hdl.handle.net/2123/2183.

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The term boset was coined by Patrick Jordan, both as an abbreviation of biordered set, and as a generalisation of poset, itself an abbreviation of partially ordered set. A boset is a set equipped with a partial multiplication and two intertwining reflexive and transitive arrow relations which satisfy certain axioms. When the arrow relations coincide the boset becomes a poset. Bosets were invented by Nambooripad (in the 1970s) who developed his own version of the theory of fundamental regular semigroups, including the classical theory of fundamental inverse semigroups using semilattices, due to Munn (in the 1960s). A semigroup is fundamental if it cannot be shrunk homomorphically without collapsing its skeleton of idempotents, which is a boset. Nambooripad constructed the maximum fundamental regular semigroup with a given boset of idempotents. Fundamental semigroups and bosets are natural candidates for basic building blocks in semigroup theory because every semigroup is a coextension of a fundamental semigroup in which the boset of idempotents is undisturbed. Recently Jordan reproved Nambooripad's results using a new construction based on arbitrary bosets. In this thesis we prove that this construction is always fundamental, which was previously known only for regular bosets, and also that it possesses a certain maximality property with respect to semigroups which are generated by regular elements. For nonregular bosets this constuction may be regular or nonregular. We introduce a class of bosets, called sawtooth bosets, which contain many regular and nonregular examples, and correct a criterion of Jordan's for the regularity of this construction for sawtooth bosets with two teeth. We also introduce a subclass, called cyclic sawtooth bosets, also containing many regular and nonregular examples, for which the construction is always regular.
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Roberts, Brad. "On bosets and fundamental semigroups." University of Sydney, 2007. http://hdl.handle.net/2123/2183.

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Doctor of Philosphy (PhD)
The term boset was coined by Patrick Jordan, both as an abbreviation of biordered set, and as a generalisation of poset, itself an abbreviation of partially ordered set. A boset is a set equipped with a partial multiplication and two intertwining reflexive and transitive arrow relations which satisfy certain axioms. When the arrow relations coincide the boset becomes a poset. Bosets were invented by Nambooripad (in the 1970s) who developed his own version of the theory of fundamental regular semigroups, including the classical theory of fundamental inverse semigroups using semilattices, due to Munn (in the 1960s). A semigroup is fundamental if it cannot be shrunk homomorphically without collapsing its skeleton of idempotents, which is a boset. Nambooripad constructed the maximum fundamental regular semigroup with a given boset of idempotents. Fundamental semigroups and bosets are natural candidates for basic building blocks in semigroup theory because every semigroup is a coextension of a fundamental semigroup in which the boset of idempotents is undisturbed. Recently Jordan reproved Nambooripad's results using a new construction based on arbitrary bosets. In this thesis we prove that this construction is always fundamental, which was previously known only for regular bosets, and also that it possesses a certain maximality property with respect to semigroups which are generated by regular elements. For nonregular bosets this constuction may be regular or nonregular. We introduce a class of bosets, called sawtooth bosets, which contain many regular and nonregular examples, and correct a criterion of Jordan's for the regularity of this construction for sawtooth bosets with two teeth. We also introduce a subclass, called cyclic sawtooth bosets, also containing many regular and nonregular examples, for which the construction is always regular.
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