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Journal articles on the topic 'Fundamental semigroup'

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Published: 4 June 2021

Last updated: 21 February 2023

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1

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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2

Wang, Shoufeng. "On generalized Ehresmann semigroups." Open Mathematics 15, no. 1 (September 6, 2017): 1132–47. http://dx.doi.org/10.1515/math-2017-0091.

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Abstract As a generalization of the class of inverse semigroups, the class of Ehresmann semigroups is introduced by Lawson and investigated by many authors extensively in the literature. In particular, Gomes and Gould construct a fundamental Ehresmann semigroup CE from a semilattice E which plays for Ehresmann semigroups the role that TE plays for inverse semigroups, where TE is the Munn semigroup of a semilattice E. From a varietal perspective, Ehresmann semigroups are derived from reduction of inverse semigroups. In this paper, from varietal perspective Ehresmann semigroups are extended to generalized Ehresmann semigroups derived instead from normal orthodox semigroups (i.e. regular semigroups whose idempotents form normal bands) with an inverse transversal. We present here a semigroup C(I,Λ,E∘) from an admissible triple (I, Λ, E∘) that plays for generalized Ehresmann semigroups the role that CE from a semilattice E plays for Ehresmann semigroups. More precisely, we show that a semigroup is a fundamental generalized Ehresmann semigroup whose admissible triple is isomorphic to (I, Λ, E∘) if and only if it is (2,1,1,1)-isomorphic to a quasi-full (2,1,1,1)-subalgebra of C(I,Λ,E∘). Our results generalize and enrich some results of Fountain, Gomes and Gould on weakly E-hedges semigroups and Ehresmann semigroups.

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3

Neeb, Karl-Hermann. "On the fundamental group of a Lie semigroup." Glasgow Mathematical Journal 34, no. 3 (September 1992): 379–94. http://dx.doi.org/10.1017/s0017089500008983.

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The simplest type of Lie semigroups are closed convex cones in finite dimensional vector spaces. In general one defines a Lie semigroup to be a closed subsemigroup of a Lie group which is generated by one-parameter semigroups. If W is a closed convex cone in a vector space V, then W is convex and therefore simply connected. A similar statement for Lie semigroups is false in general. There exist generating Lie semigroups in simply connected Lie groups which are not simply connected (Example 1.15). To find criteria for cases when this is true, one has to consider the homomorphisminduced by the inclusion mapping i:S→G, where S is a generating Lie semigroup in the Lie group G. Our main results concern the description of the image and the kernel of this mapping. We show that the image is the fundamental group of the largest covering group of G, into which S lifts, and that the kernel is the fundamental group of the inverse image of 5 in the universal covering group G. To get these results we construct a universal covering semigroup S of S. If j: H(S): = S ∩ S-1 →S is the inclusion mapping of the unit group of S into S, then it turns out that the kernel of the induced mappingmay be identfied with the fundamental group of the unit group H(S)of S and that its image corresponds to the intersection H(S)0 ⋂π1(S), where π1(s) is identified with a central subgroup of S.

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4

Miller, Craig, and Nik Ruškuc. "Right noetherian semigroups." International Journal of Algebra and Computation 30, no. 01 (September 25, 2019): 13–48. http://dx.doi.org/10.1142/s0218196719500632.

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A semigroup [Formula: see text] is right noetherian if every right congruence on [Formula: see text] is finitely generated. In this paper, we present some fundamental properties of right noetherian semigroups, discuss how semigroups relate to their substructures with regard to the property of being right noetherian, and investigate whether this property is preserved under various semigroup constructions.

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5

Urlu Özalan, Nurten, A. Sinan Çevik, and Eylem Güzel Karpuz. "A new semigroup obtained via known ones." Asian-European Journal of Mathematics 12, no. 06 (October 14, 2019): 2040008. http://dx.doi.org/10.1142/s1793557120400082.

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The goal of this paper is to establish a new class of semigroups based on both Rees matrix and completely [Formula: see text]-simple semigroups. We further present some fundamental properties and finiteness conditions for this new semigroup structure.

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6

Edwards, P. M. "Fundamental semigroups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 99, no. 3-4 (1985): 313–17. http://dx.doi.org/10.1017/s0308210500014323.

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SynopsisAn idempotent-separating congruence μ is studied further in this paper. It is shown to satisfy special properties with respect to regular elements and to group-bound elements. It is shown that for any semigroup S, μ is the identity congruence on S/μ. From this, it can be shown that S/μ is fundamental for any semigroup S. Some alternative characterizations of μ are given and applied to yield sufficient conditions for a subsemigroup T of S to satisfy μ (T) = μ (S) ∩ (T × T), whence T is fundamental if S is fundamental.

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7

Habib, Sana, Harish Garg, Yufeng Nie, and Faiz Muhammad Khan. "An Innovative Approach towards Possibility Fuzzy Soft Ordered Semigroups for Ideals and Its Application." Mathematics 7, no. 12 (December 3, 2019): 1183. http://dx.doi.org/10.3390/math7121183.

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The objective of this paper is put forward the novel concept of possibility fuzzy soft ideals and the possibility of fuzzy soft interior ideals. The various results in the form of the theorems with these notions are presented and further validated by suitable examples. In modern life decision-making problems, there is a wide applicability of the possibility fuzzy soft ordered semigroup which has also been constructed in the paper to solve the decision-making process. Elementary and fundamental concepts including regular, intra-regular and simple ordered semigroups in terms of possibility fuzzy soft ordered semigroup are presented. Later, the concept of left (resp. right) regular and left (resp. right) simple in terms of possibility fuzzy soft ordered semigroups are delivered. Finally, the notion of possibility fuzzy soft semiprime ideals in an ordered semigroup is defined and illustrated by theorems and example.

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8

Simard, Arnaud. "Counterexamples concerning powers of sectorial operators on a Hilbert space." Bulletin of the Australian Mathematical Society 60, no. 3 (December 1999): 459–68. http://dx.doi.org/10.1017/s0004972700036613.

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We give explicit constructions of semigroups and operators with particular properties. First we build a bounded C0-semigroup which is invertible and which is not similar to a semigroup of contractions. Afterwards we exhibit operators which admit bounded imaginary powers of angle ω > 0 on a Hilbert space but which do not admit a bounded functional calculus on the sector of angle ω. (This gives the limit of McIntosh's fundamental result.) Finally we build, in the 2-dimensional Hilbert space, an operator which is not the negative generator of a semigroup of contractions, although its imaginary powers are bounded by eπ|s|/2.

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9

Goberstein, Simon M. "Inverse semigroups with certain types of partial automorphism monoids." Glasgow Mathematical Journal 32, no. 2 (May 1990): 189–95. http://dx.doi.org/10.1017/s0017089500009204.

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AbstractFor an inverse semigroup S, the set of all isomorphisms betweeninverse subsemigroups of S is an inverse monoid under composition which is denoted by (S) and called the partial automorphism monoid of S. Kirkwood [7] and Libih [8] determined which groups have Clifford partial automorphism monoids. Here we investigate the structure of inverse semigroups whose partial automorphism monoids belong to certain other important classes of inverse semigroups. First of all, we describe (modulo so called “exceptional” groups) all inverse semigroups S such that (S) is completely semisimple. Secondly, for an inverse semigroup S, we find a convenient description of the greatest idempotent-separating congruence on (S), using a well-known general expression for this congruence due to Howie, and describe all those inverse semigroups whose partial automorphism monoids are fundamental.

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10

van Gool, Samuel J., and Benjamin Steinberg. "Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes." Canadian Mathematical Bulletin 62, no. 1 (January 7, 2019): 199–208. http://dx.doi.org/10.4153/cmb-2018-014-8.

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AbstractThis paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell’s aperiodic pointlike theorem. We use a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided semidirect product whose components are built from two subsemigroups $T_{1}$, $T_{2}$, which together generate $T$, and the subsemigroup generated by their setwise product $T_{1}T_{2}$. In this sense we decompose $T$ by merging the subsemigroups $T_{1}$ and $T_{2}$. More generally, our technique merges semigroup homomorphisms from free semigroups.

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11

Al-Masarwah, Anas, Mohammed Alqahtani, and Majdoleen Abu Qamar. "Groups and Structures of Commutative Semigroups in the Context of Cubic Multi-Polar Structures." Symmetry 14, no. 7 (July 21, 2022): 1493. http://dx.doi.org/10.3390/sym14071493.

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In recent years, the m-polar fuzziness structure and the cubic structure have piqued the interest of researchers and have been commonly implemented in algebraic structures like groupoids, semigroups, groups, rings and lattices. The cubic m-polar (CmP) structure is a generalization of m-polar fuzziness and cubic structures. The intent of this research is to extend the CmP structures to the theory of groups and semigroups. In the present research, we preface the concept of the CmP groups and probe many of its characteristics. This concept allows the membership grade and non-membership grade sequence to have a set of m-tuple interval-valued real values and a set of m-tuple real values between zero and one. This new notation of group (semigroup) serves as a bridge among CmP structure, classical set and group (semigroup) theory and also shows the effect of the CmP structure on a group (semigroup) structure. Moreover, we derive some fundamental properties of CmP groups and support them by illustrative examples. Lastly, we vividly construct semigroup and groupoid structures by providing binary operations for the CmP structure and provide some dominant properties of these structures.

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12

Liu, Haijun, and Xiaojiang Guo. "The lattice of (2, 1)-congruences on a left restriction semigroup." Open Mathematics 20, no. 1 (January 1, 2022): 1159–72. http://dx.doi.org/10.1515/math-2022-0492.

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Abstract All the (2, 1)-congruences on a left restriction semigroup become a complete sublattice of its lattice of congruences. The aim of this article is to study certain fundamental properties of this complete sublattice. We introduce k k -, K K -, t t -, and T T -operators on this sublattice and obtain some properties. As applications, the remarkable (2, 1)-congruences are characterized. These results extend the corresponding results on inverse semigroups and ample semigroups.

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13

LAWSON, M. V. "NON-COMMUTATIVE STONE DUALITY: INVERSE SEMIGROUPS, TOPOLOGICAL GROUPOIDS AND C*-ALGEBRAS." International Journal of Algebra and Computation 22, no. 06 (August 31, 2012): 1250058. http://dx.doi.org/10.1142/s0218196712500580.

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We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse ∧-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse ∧-semigroups arise as completions of inverse ∧-semigroups we call pre-Boolean. An inverse ∧-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where tight filters are defined by combining ideas of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson–Higman groups Gn, r. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz–Krieger C*-algebras. An elementary application of our theory shows that the finite, fundamental Boolean inverse ∧-semigroups are just the finite direct products of finite symmetric inverse monoids. Finally, we explain how tight filters are related to prime filters setting the scene for future work.

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14

Kilibarda, Vesna. "On the Algebra of Semigroup Diagrams." International Journal of Algebra and Computation 07, no. 03 (June 1997): 313–38. http://dx.doi.org/10.1142/s0218196797000150.

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In this work we enrich the geometric method of semigroup diagrams to study semigroup presentations. We introduce a process of reduction on semigroup diagrams which leads to a natural way of multiplying semigroup diagrams associated with a given semigroup presentation. With respect to this multiplication the set of reduced semigroup diagrams is a groupoid. The main result is that the groupoid [Formula: see text] of reduced semigroup diagrams over the presentation S = <X:R> may be identified with the fundamental groupoid γ (KS) of a certain 2-dimensional complex KS. Consequently, the vertex groups of the groupoid [Formula: see text] are isomorphic to the fundamental groups of the complex KS. The complex we discovered was first considered in the paper of Craig Squier, published only recently. Steven Pride has also independently defined a 2-dimensional complex isomorphic to KS in relation to his work on low-dimensional homotopy theory for monoids. Some structural information about the fundamental groups of the complex KS are presented. The class of these groups contains all finitely generated free groups and is closed under finite direct and finite free products. Many additional results on the structure of these groups may be found in the paper of Victor Guba and Mark Sapir.

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15

Grillet, Pierre Antoine. "On the fundamental double four-spiral semigroup." Bulletin of the Belgian Mathematical Society - Simon Stevin 3, no. 2 (1996): 201–8. http://dx.doi.org/10.36045/bbms/1105540792.

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16

Wang, Feng-Yu. "Semigroup properties for the second fundamental form." Documenta Mathematica 15 (2010): 527–43. http://dx.doi.org/10.4171/dm/305.

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17

DAVVAZ, B., W. A. DUDEK, and S. MIRVAKILI. "NEUTRAL ELEMENTS, FUNDAMENTAL RELATIONS AND n-ARY HYPERSEMIGROUPS." International Journal of Algebra and Computation 19, no. 04 (June 2009): 567–83. http://dx.doi.org/10.1142/s0218196709005226.

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The main tools in the theory of n-ary hyperstructures are the fundamental relations. The fundamental relation on an n-ary hypersemigroup is defined as the smallest equivalence relation so that the quotient would be the n-ary semigroup. In this paper we study neutral elements in n-ary hypersemigroups and introduce a new strongly compatible equivalence relation on an n-ary hypersemigroup so that the quotient is a commutative n-ary semigroup. Also we determine some necessary and sufficient conditions so that this relation is transitive. Finally, we prove that this relation is transitive on an n-ary hypergroup with neutral (identity) element.

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18

Gigoń, Roman S. "Certain congruences on eventually regular semigroups I." Studia Scientiarum Mathematicarum Hungarica 52, no. 4 (December 2015): 434–49. http://dx.doi.org/10.1556/012.2015.52.4.1313.

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A semigroup is called eventually regular if each of its elements has a regular power. In this paper we study certain fundamental congruences on an eventually regular semigroup. We generalize some results of Howie and Lallement (1966) and LaTorre (1983). In particular, we give a full description of the semilattice of group congruences (together with the least such a congruence) on an arbitrary eventually regular (orthodox) semigroup. Moreover, we investigate UBG-congruences on an eventually regular semigroup. Finally, we study the eventually regular subdirect products of an E-unitary semigroup and a Clifford semigroup.

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19

SKEIDE, MICHAEL. "A SIMPLE PROOF OF THE FUNDAMENTAL THEOREM ABOUT ARVESON SYSTEMS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 09, no. 02 (June 2006): 305–14. http://dx.doi.org/10.1142/s0219025706002378.

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With every E0-semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E0-semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and allows for a generalization to product systems of Hilbert module (to be published elsewhere).

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20

BALLOT, CHRISTIAN, and MIREILLE CAR. "ON MURATA DENSITIES." International Journal of Number Theory 07, no. 07 (November 2011): 1717–36. http://dx.doi.org/10.1142/s179304211100440x.

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In this paper, we set up an abstract theory of Murata densities, well tailored to general arithmetical semigroups. In [On certain densities of sets of primes, Proc. Japan Acad. Ser. A Math. Sci.56(7) (1980) 351–353; On some fundamental relations among certain asymptotic densities, Math. Rep. Toyama Univ.4(2) (1981) 47–61], Murata classified certain prime density functions in the case of the arithmetical semigroup of natural numbers. Here, it is shown that the same density functions will obey a very similar classification in any arithmetical semigroup whose sequence of norms satisfies certain general growth conditions. In particular, this classification holds for the set of monic polynomials in one indeterminate over a finite field, or for the set of ideals of the ring of S-integers of a global function field (S finite).

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21

Hooshmand, M. H. "Upper and Lower Periodic Subsets of Semigroups." Algebra Colloquium 18, no. 03 (September 2011): 447–60. http://dx.doi.org/10.1142/s1005386711000332.

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In this paper, a new topic about a vast class of subsets of semigroups and binary systems, which contains all ideals, periodic subsets and sub-semigroups, is introduced and studied. In fact, the “upper periodic subsets” can be considered as a generalization of the conception “ideals”. We prove a fundamental theorem which states that if A is a (left) upper B-periodic subset of a semigroup S, then under some conditions, it has a unique direct representation [Formula: see text], where B1=B ∪ {1} and B ⊆ 𝔅 ≤ S. Especially, we prove a unique direct representation for upper and lower T-periodic subsets, and classify all sub-semigroups of S containing a fixed element T to three classes. This classification gives us more interesting properties for the real semigroups. At last, we characterize upper and lower T-periodic subsets of semigroups and groups.

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22

Easdown, D. "The minimal faithful degree of a fundamental inverse semigroup." Bulletin of the Australian Mathematical Society 35, no. 3 (June 1987): 373–78. http://dx.doi.org/10.1017/s0004972700013356.

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This paper shows that the smallest size of a set for which a finite fundamental inverse semigroup can be faithfully represented by partial transformations of that set is the number of join irreducible elements of its semilattice of idempotents.

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23

SCHWAB, EMIL DANIEL, and PENTTI HAUKKANEN. "A UNIQUE FACTORIZATION IN COMMUTATIVE MÖBIUS MONOIDS." International Journal of Number Theory 04, no. 04 (August 2008): 549–61. http://dx.doi.org/10.1142/s1793042108001523.

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We show that any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Particular attention is paid to standard examples, which arise from the bicyclic semigroup and the multiplicative analogue of the bicyclic semigroup. The second example shows that the Fundamental Theorem of Arithmetic is a special case of the unique factorization theorem in commutative Möbius monoids. As an application, we study generalized arithmetical functions defined on an arbitrary commutative Möbius monoid.

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24

Nambooripad, K. S. S., and F. J. Pastijn. "The fundamental representation of a strongly regular Baer semigroup." Journal of Algebra 92, no. 2 (February 1985): 283–302. http://dx.doi.org/10.1016/0021-8693(85)90121-8.

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25

Rehman, Ubaid Ur, Tahir Mahmood, and Muhammad Naeem. "Bipolar complex fuzzy semigroups." AIMS Mathematics 8, no. 2 (2022): 3997–4021. http://dx.doi.org/10.3934/math.2023200.

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<abstract> <p>The notion of the bipolar complex fuzzy set (BCFS) is a fundamental notion to be considered for tackling tricky and intricate information. Here, in this study, we want to expand the notion of BCFS by giving a general algebraic structure for tackling bipolar complex fuzzy (BCF) data by fusing the conception of BCFS and semigroup. Firstly, we investigate the bipolar complex fuzzy (BCF) sub-semigroups, BCF left ideal (BCFLI), BCF right ideal (BCFRI), BCF two-sided ideal (BCFTSI) over semigroups. We also introduce bipolar complex characteristic function, positive $ \left(\omega , \eta \right) $-cut, negative $ \left(\varrho , \sigma \right) $-cut, positive and $ \left(\left(\omega , \eta \right), \left(\varrho , \sigma \right)\right) $-cut. Further, we study the algebraic structure of semigroups by employing the most significant concept of BCF set theory. Also, we investigate numerous classes of semigroups such as right regular, left regular, intra-regular, and semi-simple, by the features of the bipolar complex fuzzy ideals. After that, these classes are interpreted concerning BCF left ideals, BCF right ideals, and BCF two-sided ideals. Thus, in this analysis, we portray that for a semigroup $ Ş $ and for each BCFLI $ {М}_{1} = \left({\mathrm{\lambda }}_{P-{М}_{1}}, {\mathrm{\lambda }}_{N-{М}_{1}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{1}}+\iota {\mathrm{\lambda }}_{IP-{М}_{1}}, {\mathrm{\lambda }}_{RN-{М}_{1}}+\iota {\mathrm{\lambda }}_{IN-{М}_{1}}\right) $ and BCFRI $ {М}_{2} = \left({\mathrm{\lambda }}_{P-{М}_{2}}, {\mathrm{\lambda }}_{N-{М}_{2}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{2}}+\iota {\mathrm{\lambda }}_{IP-{М}_{2}}, {\mathrm{\lambda }}_{RN-{М}_{2}}+\iota {\mathrm{\lambda }}_{IN-{М}_{2}}\right) $ over $ Ş $, $ {М}_{1}\cap {М}_{2} = {М}_{1}⊚{М}_{2} $ if and only if $ Ş $ is a regular semigroup. At last, we introduce regular, intra-regular semigroups and show that $ {М}_{1}\cap {М}_{2}\preccurlyeq {М}_{1}⊚{М}_{2} $ for each BCFLI $ {М}_{1} = \left({\mathrm{\lambda }}_{P-{М}_{1}}, {\mathrm{\lambda }}_{N-{М}_{1}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{1}}+\iota {\mathrm{\lambda }}_{IP-{М}_{1}}, {\mathrm{\lambda }}_{RN-{М}_{1}}+\iota {\mathrm{\lambda }}_{IN-{М}_{1}}\right) $ and for each BCFRI $ {М}_{2} = \left({\mathrm{\lambda }}_{P-{М}_{2}}, {\mathrm{\lambda }}_{N-{М}_{2}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{2}}+\iota {\mathrm{\lambda }}_{IP-{М}_{2}}, {\mathrm{\lambda }}_{RN-{М}_{2}}+\iota {\mathrm{\lambda }}_{IN-{М}_{2}}\right) $ over $ Ş $ if and only if a semigroup $ Ş $ is regular and intra-regular.</p> </abstract>

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26

Goberstein, Simon M. "Inverse semigroups determined by their partial automorphism monoids." Journal of the Australian Mathematical Society 81, no. 2 (October 2006): 185–98. http://dx.doi.org/10.1017/s1446788700015810.

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AbstractThe partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup S with no isolated nontrivial subgroups is lattice determined ‘modulo semilattices’ and if T is an inverse semigroup whose partial automorphism monoid is isomorphic to that of S, then either S and T are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if T is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of S and T, respectively, are isomorphic. Moreover, for these results to hold, the conditions that S be tightly connected and have no isolated nontrivial subgroups are essential.

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27

Bennett, Paul. "On the Structure of Inverse Semigroup Amalgams." International Journal of Algebra and Computation 07, no. 05 (October 1997): 577–604. http://dx.doi.org/10.1142/s0218196797000265.

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This paper is the second of two papers devoted to the study of amalgamated free products of inverse semigroups. We use the characterization of the Schützenberger automata given previously by the author to obtain structural results and preservational properties of lower bounded amalgams. Haataja, Margolis and Meakin have shown that if [S1,S2;U is an amalgam of regular semigroups in which S1∩ S2=U is a full regular subsemigroup of S1 and S2, then the maximal subgroups of the amalgamated free product S1*U S2 may be described by the fundamental groups of certain bipartite graphs of groups. In this paper we show that the maximal subgroups of a lower bounded amalgam [S1,S2;U] are either isomorphic copies of subgroups of S1 and S2 or can be described by the same Bass-Serre theory characterization. It follows, as for the regular case, that if S1 and S2 are combinatorial, then the maximal subgroups of S1*U S2 are free. By studying the endomorphism monoids of the Schützenberger graphs we obtain a number of results concerning when inverse semigroup properties are preserved under the amalgamated free product construction. For example, necessary and sufficient conditions are given for S1*U S2 to be completely semisimple. Under a mild assumption we establish necessary and sufficient conditions for S1*U S2 to have finite ℛ-classes. This enables us to reprove a result of Cherubini, Meakin and Piochi on amalgams of free inverse semigroups. Finally we give sufficient conditions for S1*U S2 to be E-unitary.

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Liang, Jin, Falun Huang, and Tijun Xiao. "Exponential stability for abstract linear autonomous functional differential equations with infinite delay." International Journal of Mathematics and Mathematical Sciences 21, no. 2 (1998): 255–59. http://dx.doi.org/10.1155/s0161171298000362.

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Based on our preceding paper, this note is concerned with the exponential stability of the solution semigroup for the abstract linear autonomous functional differential equationx˙(t)=L(xt) (∗)whereLis a continuous linear operator on some abstract phase spaceBinto a Banach spaceE. We prove that the solution semigroup of(∗)is exponentially stable if and only if the fundamental operator(∗)is exponentially stable and the phase spaceBis an exponentially fading memory space.

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CORNWELL, CHRISTOPHER R., and STEPHEN P. HUMPHRIES. "COUNTING FUNDAMENTAL PATHS IN CERTAIN GARSIDE SEMIGROUPS." Journal of Knot Theory and Its Ramifications 17, no. 02 (February 2008): 191–211. http://dx.doi.org/10.1142/s0218216508006051.

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For elements a, b of a monoid, define the word pk(a,b) = abab⋯ of length k. We find the number of words in a, b which are equal to pk(a,b)n in the Artin semigroup < a,b|pk(a,b) = pk(b,a) >. This number is related to counting certain paths in the ℕ × ℕ lattice. These Artin groups are examples of two generator Garside groups. We also define other examples of Garside groups G on more than two generators, having fundamental word Δ, and similarly find the number of words equal in G to Δn.

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30

Abdeljawad, Thabet, and Arran Fernandez. "On a New Class of Fractional Difference-Sum Operators with Discrete Mittag-Leffler Kernels." Mathematics 7, no. 9 (August 22, 2019): 772. http://dx.doi.org/10.3390/math7090772.

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We formulate a new class of fractional difference and sum operators, study their fundamental properties, and find their discrete Laplace transforms. The method depends on iterating the fractional sum operators corresponding to fractional differences with discrete Mittag–Leffler kernels. The iteration process depends on the binomial theorem. We note in particular the fact that the iterated fractional sums have a certain semigroup property, and hence, the new introduced iterated fractional difference-sum operators have this semigroup property as well.

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31

Yamamura, Akihiro. "Locally full HNN extensions of inverse semigroups." Journal of the Australian Mathematical Society 70, no. 2 (April 2001): 235–72. http://dx.doi.org/10.1017/s1446788700002639.

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AbstractWe investigate a locally full HNN extension of an inverse semigroup. A normal form theorem is obtained and applied to the word problem. We construct a tree and show that a maximal subgroup of a locally full HNN extension acts on the tree without inversion. Bass-Serre theory is employed to obtain a group presentation of the maximal subgroup as a fundamental group of a certain graph of groups associated with the D-structure of the original semigroup.

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32

Shevchuk, R. V., I. Ya Savka, and Z. M. Nytrebych. "The nonlocal boundary value problem for one-dimensional backward Kolmogorov equation and associated semigroup." Carpathian Mathematical Publications 11, no. 2 (December 31, 2019): 463–74. http://dx.doi.org/10.15330/cmp.11.2.463-474.

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This paper is devoted to a partial differential equation approach to the problem of construction of Feller semigroups associated with one-dimensional diffusion processes with boundary conditions in theory of stochastic processes. In this paper we investigate the boundary-value problem for a one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) in curvilinear bounded domain with one of the variants of nonlocal Feller-Wentzell boundary condition. We restrict our attention to the case when the boundary condition has only one term and it is of the integral type. The classical solution of the last problem is obtained by the boundary integral equation method with the use of the fundamental solution of backward Kolmogorov equation and the associated parabolic potentials. This solution is used to construct the Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle leaves the boundary of the domain by jumps.

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33

Mirvakili, S., S. M. Anvariyeh, and B. Davvaz. "Γ-Semihypergroups and Regular Relations." Journal of Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/915250.

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In this paper we discuss the structure ofΓ-semihypergroups. We prove some basic results and present several examples ofΓ-semihypergroups. Also, we obtain some properties of regular and strongly regular relations on aΓ-semihypergroup and construct aΓ-semigroup from aΓ-semihypergroup by using the notion of fundamental relation.

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34

BELL, PAUL C., and IGOR POTAPOV. "ON THE UNDECIDABILITY OF THE IDENTITY CORRESPONDENCE PROBLEM AND ITS APPLICATIONS FOR WORD AND MATRIX SEMIGROUPS." International Journal of Foundations of Computer Science 21, no. 06 (December 2010): 963–78. http://dx.doi.org/10.1142/s0129054110007660.

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In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of integral square matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several questions for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words.

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35

Hamana, Masamichi. "Partial *-Automorphisms, Normalizers, and Submodules in Monotone Complete C*-Algebras." Canadian Journal of Mathematics 58, no. 6 (December 1, 2006): 1144–202. http://dx.doi.org/10.4153/cjm-2006-042-0.

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AbstractFor monotone complete C*-algebras A ⊂ B with A contained in B as a monotone closed C*-subalgebra, the relation X = AsA gives a bijection between the set of all monotone closed linear subspaces X of B such that AX + XA ⊂ X and XX* + X*X ⊂ A and a set of certain partial isometries s in the “normalizer” of A in B, and similarly for the map s ⟼ Ad s between the latter set and a set of certain “partial *-automorphisms” of A. We introduce natural inverse semigroup structures in the set of such X's and the set of partial *-automorphisms of A, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough B the homomorphism becomes surjective and all the partial *-automorphisms of A are realized via partial isometries in B. In particular, the inverse semigroup associated with a type II1 von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the C*-algebra version of these results.

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36

Gulistan, Muhammad, Feng Feng, Madad Khan, and Aslıhan Sezgin. "Characterizations of Right Weakly Regular Semigroups in Terms of Generalized Cubic Soft Sets." Mathematics 6, no. 12 (November 30, 2018): 293. http://dx.doi.org/10.3390/math6120293.

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Cubic sets are the very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of [ 0 , 1 ] and a number from [ 0 , 1 ] . Generalized cubic sets generalized the cubic sets with the help of cubic point. On the other hand Soft sets were proved to be very effective tool for handling imprecision. Semigroups are the associative structures have many applications in the theory of Automata. In this paper we blend the idea of cubic sets, generalized cubic sets and semigroups with the soft sets in order to develop a generalized approach namely generalized cubic soft sets in semigroups. As the ideal theory play a fundamental role in algebraic structures through this we can make a quotient structures. So we apply the idea of neutrosophic cubic soft sets in a very particular class of semigroups namely weakly regular semigroups and characterize it through different types of ideals. By using generalized cubic soft sets we define different types of generalized cubic soft ideals in semigroups through three different ways. We discuss a relationship between the generalized cubic soft ideals and characteristic functions and cubic level sets after providing some basic operations. We discuss two different lattice structures in semigroups and show that in the case when a semigroup is regular both structures coincides with each other. We characterize right weakly regular semigroups using different types of generalized cubic soft ideals. In this characterization we use some classical results as without them we cannot prove the inter relationship between a weakly regular semigroups and generalized cubic soft ideals. This generalization leads us to a new research direction in algebraic structures and in decision making theory.

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37

Li, Pengtao, Tao Qian, Zhiyong Wang, and Chao Zhang. "Regularity of Fractional Heat Semigroup Associated with Schrödinger Operators." Fractal and Fractional 6, no. 2 (February 14, 2022): 112. http://dx.doi.org/10.3390/fractalfract6020112.

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Let L=−Δ+V be a Schrödinger operator, where the potential V belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup {e−tLα}t>0, 0<α<1, associated with L. By the aid of the fundamental solution of the heat equation: ∂tu+Lu=∂tu−Δu+Vu=0, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel Kα,tL(·,·), respectively. This method is independent of the Fourier transform, and can be applied to the second-order differential operators whose heat kernels satisfy the Gaussian upper bounds. As an application, we establish a Carleson measure characterization of the Campanato-type space BMOLγ(Rn) via the fractional heat semigroup {e−tLα}t>0.

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38

Sivasankar, Sivajiganesan, Ramalingam Udhayakumar, Velmurugan Subramanian, Ghada AlNemer, and Ahmed M. Elshenhab. "Existence of Hilfer Fractional Stochastic Differential Equations with Nonlocal Conditions and Delay via Almost Sectorial Operators." Mathematics 10, no. 22 (November 21, 2022): 4392. http://dx.doi.org/10.3390/math10224392.

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In this article, we examine the existence of Hilfer fractional (HF) stochastic differential systems with nonlocal conditions and delay via almost sectorial operators. The major methods depend on the semigroup of operators method and the Mo¨nch fixed-point technique via the measure of noncompactness, and the fundamental theory of fractional calculus. Finally, to clarify our key points, we provide an application.

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39

Shevchuk, R., and I. Savka. "THE NONLOCAL CONJUGATION PROBLEM FOR A LINEAR SECOND ORDER PARABOLIC EQUATION OF KOLMOGOROV'S TYPE WITH DISCONTINUOUS COEFFICIENTS." Bukovinian Mathematical Journal 10, no. 2 (2022): 249–64. http://dx.doi.org/10.31861/bmj2022.02.20.

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In this paper, we construct the two-parameter Feller semigroup associated with a certain one-dimensional inhomogeneous Markov process. This process may be described as follows. At the interior points of the finite number of intervals $(-\infty,r_1(s)),\,(r_1(s),r_2(s)),\ldots,\,(r_{n}(s),\infty)$ separated by points $r_i(s)\,(i=1,\ldots,n)$, the positions of which depend on the time variable, this process coincides with the ordinary diffusions given there by their generating differential operators, and its behavior on the common boundaries of these intervals is determined by the Feller-Wentzell conjugation conditions of the integral type, each of which corresponds to the inward jump phenomenon from the boundary. The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding nonlocal conjugation problem for a second order linear parabolic equation of Kolmogorov’s type with discontinuous coefficients. The main part of the paper consists in the investigation of this parabolic conjugation problem, the peculiarity of which is that the domains on the plane, where the equations are given, are curvilinear and have non-smooth boundaries: the functions $r_i(s)\,(i=1,\ldots,n)$, which determine the boundaries of these domains satisfy only the Hölder condition with exponent greater than $\frac{1}{2}$. Its classical solvability in the space of continuous functions is established by the boundary integral equations method with the use of the fundamental solutions of the uniformly parabolic equations and the associated potentials. It is also proved that the solution of this problem has a semigroup property. The availability of the integral representation for the constructed semigroup allows us to prove relatively easily that this semigroup yields the Markov process.

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40

Karthikeyan, Kulandhaivel, Amar Debbouche, and Delfim F. M. Torres. "Analysis of Hilfer Fractional Integro-Differential Equations with Almost Sectorial Operators." Fractal and Fractional 5, no. 1 (March 8, 2021): 22. http://dx.doi.org/10.3390/fractalfract5010022.

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In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory.

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41

MARINELLI, CARLO, and MICHAEL RÖCKNER. "ON UNIQUENESS OF MILD SOLUTIONS FOR DISSIPATIVE STOCHASTIC EVOLUTION EQUATIONS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 13, no. 03 (September 2010): 363–76. http://dx.doi.org/10.1142/s0219025710004152.

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In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption.

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42

Tajmouati, Abdelaziz, and Hamid Boua. "Spectral Mapping Theorem for C0-Semigroups of Drazin spectrum." Boletim da Sociedade Paranaense de Matemática 38, no. 3 (February 18, 2019): 63–69. http://dx.doi.org/10.5269/bspm.v38i3.38404.

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Let $(T(t))_{t\geq 0}$ be a $C_0$ semigroup of bounded linear operators on a Banach space $X$ and denote its generator by $A$. A fundamental problem to decide whether the Drazin spectrum of each operator $T(t)$ can be obtained from the Drazin spectrum of $A$. In particular, one hopes that the Drazin Spectral Mapping Theorem holds, i.e., $e^{t \sigma_{D}(A)}=\sigma_{D}(T(t))\backslash \{0\}$ for all $t \geq 0$.

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43

Liang, Jin, and Tijun Xiao. "Functional differential equations with infinite delay in Banach spaces." International Journal of Mathematics and Mathematical Sciences 14, no. 3 (1991): 497–508. http://dx.doi.org/10.1155/s0161171291000686.

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In this paper, a definition of the fundamental operator for the linear autonomous functional differential equation with infinite delay in a Banach space is given, and some sufficient and necessary conditions of the fundamental operator being exponentially stable in abstract phase spaces which satisfy some suitable hypotheses are obtained. Moreover, we discuss the relation between the exponential asymptotic stability of the zero solution of nonlinear functional differential equation with infinite delay in a Banach space and the exponential stability of the solution semigroup of the corresponding linear equation, and find that the exponential stability problem of the zero solution for the nonlinear equation can be discussed only in the exponentially fading memory phase space.

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44

Chen, Mu-Fa. "On three classical problems for Markov chains with continuous time parameters." Journal of Applied Probability 28, no. 2 (June 1991): 305–20. http://dx.doi.org/10.2307/3214868.

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For a given transition rate, i.e., a Q-matrix Q = (qij) on a countable state space, the uniqueness of the Q-semigroup P(t) = (Pij(t)), the recurrence and the positive recurrence of the corresponding Markov chain are three fundamental and classical problems, treated in many textbooks. As an addition, this paper introduces some practical results motivated from the study of a type of interacting particle systems, reaction diffusion processes. The main results are theorems (1.11), (1.17) and (1.18). Their proofs are quite straightforward.

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45

Farshi, Mehdi, Bijan Davvaz, and Saeed Mirvakili. "On the g-hypergroupoids associated with g-hypergraphs." Filomat 31, no. 15 (2017): 4819–31. http://dx.doi.org/10.2298/fil1715819f.

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In this paper, we associate a partial g-hypergroupoid with a given g-hypergraph and analyze the properties of this hyperstructure. We prove that a g-hypergroupoid may be a commutative hypergroup without being a join space. Next, we define diagonal direct product of g-hypergroupoids. Further, we construct a sequence of g-hypergroupoids and investigate some relationships between it?s terms. Also, we study the quotient of a g-hypergroupoid by defining a regular relation. Finally, we describe fundamental relation of an Hv-semigroup as a g-hypergroupoid.

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46

Mehdi Ebrahimi, M., Mojgan Mahmoudi, and Mahdieh Yavari. "Absolute retractness of automata on directed complete posets." Journal of Algebra and Its Applications 16, no. 01 (January 2017): 1750008. http://dx.doi.org/10.1142/s0219498817500086.

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The notion of retractness, which is about having left inverses (reflection) for monomorphisms, is crucial in most branches of mathematics. One very important notion related to it is injectivity, which is about extending morphisms to larger domains and plays a fundamental role in many areas of mathematics as well as in computer science, under the name of complete or partial objects. Absolute retractness is tightly related to injectivity and is in fact equivalent to it in many categories. In this paper, combining the two important notions of actions of semigroups and directed complete posets, which are both crucial abstraction and useful in mathematics as well as in computer science, we consider the category Dcpo-[Formula: see text] of actions of a directed complete semigroup on directed complete posets, and study absolute retractness with respect to both monomorphisms and embeddings in this category. Among other things, we show that absolute retract ([Formula: see text]-)dcpo’s are complete but the converse is not necessarily true. Investigating the converse, we find that if we add the property of being a countable chain to completeness, over some kinds of dcpo-monoids such as dcpo-groups and commutative monoids, we get absolute retractness. Furthermore, we show that there are absolute retract [Formula: see text]-dcpo’s, which are not chains.

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47

Khan, Faiz Muhammad, Violeta Leoreanu-Fotea, Saifullah, and Amanullah. "A Benchmark Generalization of Fuzzy Soft Ideals in Ordered Semigroups." Analele Universitatii "Ovidius" Constanta - Seria Matematica 29, no. 2 (June 1, 2021): 155–71. http://dx.doi.org/10.2478/auom-2021-0023.

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Abstract In real life, variability and inaccuracy are always presentand must be calculated by either possibilistic, probabilistic, polymorphic or other uncertainty approach. This benchmark study is about to construct new types of fuzzy soft ideals i.e., (∈, ∈ ∨qk )-FSR(L)Is of ordered semigroup(OSG). Based on this inception, fuzzy soft level subsets are defined which link ordinary ideals with (∈, ∈ ∨qk )-fuzzy soft left(right) ideals. Some binary operations like ◦ λ , intersection ∩ λ and union of fuzzy soft sets ∪ λ are given and various fundamental results of ideal theory are developed through these types of fuzzy soft ideals.

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48

Derech, V. D. "Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental." Ukrainian Mathematical Journal 61, no. 1 (January 2009): 57–70. http://dx.doi.org/10.1007/s11253-009-0198-9.

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49

Derech, V. D. "Structure of a finite inverse semigroup with zero every stable order on which is fundamental or antifundamental." Ukrainian Mathematical Journal 62, no. 1 (August 2010): 31–42. http://dx.doi.org/10.1007/s11253-010-0331-9.

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50

Oprzędkiewicz, Krzysztof, and Wojciech Mitkowski. "A Memory–Efficient Noninteger–Order Discrete–Time State–Space Model of a Heat Transfer Process." International Journal of Applied Mathematics and Computer Science 28, no. 4 (December 1, 2018): 649–59. http://dx.doi.org/10.2478/amcs-2018-0050.

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Abstract A new, state space, discrete-time, and memory-efficient model of a one-dimensional heat transfer process is proposed. The model is derived directly from a time-continuous, state-space semigroup one. Its discrete version is obtained via a continuous fraction expansion method applied to the solution of the state equation. Fundamental properties of the proposed model, such as decomposition, stability, accuracy and convergence, are also discussed. Results of experiments show that the model yields good accuracy in the sense of the mean square error, and its size is significantly smaller than that of the model employing the well-known power series expansion approximation.

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