Topic
Bicyclic semigroup
About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.
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TL;DR: The planar modular partition monoid as mentioned in this paper was introduced and examined for parameters m, k ∈ Z > 0, where m is the number of columns in the planar graph.
Abstract: The primary contribution of this thesis is to introduce and examine the planar modular partition monoid
for parameters m, k ∈ Z>0, which has simultaneously and independently generated interest from other
researchers as outlined within.
Our collective understanding of related monoids, in particular the Jones, Brauer and partition monoids,
along with the algebras they generate, has heavily infuenced the direction of research by a significant
number of mathematicians and physicists. Examples include Schur-Weyl type dualities in representation
theory along with Potts, ice-type and Andrew-Baxter-Forrester models from statistical mechanics, giving
strong motivation for the planar modular partition monoid to be examined.
The original results contained within this thesis relating to the planar modular partition monoid are: the
establishment of generators; recurrence relations for the cardinality of the monoid; recurrence relations
for the cardinality of Green's R, L and D relations; and a conjecture on relations that appear to present
the planar modular partition monoid when m = 2. For diagram semigroups that are closed under vertical
reflections, characterisations of Green's R, L and H relations have previously been established using the
upper and lower patterns of bipartitions. We give a characterisation of Green's D relation with a similar
flavour for diagram semigroups that are closed under vertical reflections.
We also give a number of analogous results for the modular partition monoid, the monoid generated by
replacing diapses with m-apses in the generators of the Jones monoid, later referred to as the m-apsis
monoid, and the join of the m-apsis monoid with the symmetric group.
A further contribution of this thesis is a reasonably comprehensive exposition of the fundamentals of
diagram semigroups, which have traditionally been approached from the representation theory side and
have since blossomed into a thriving area of research in their own right.
4 citations
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TL;DR: The following problem was posed by the second author in 'Semigroup Forum' as discussed by the authors, Vol. 1, No. 1, No. 2, 1970, p. 91:
Abstract: The following problem was posed by the second author in ‘Semigroup Forum’, Vol. 1, No. 1, 1970, p. 91:
4 citations
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TL;DR: In this paper, the sum σ(e)=e+∑(−1)KeL1⋯eiK, where ei,...,em are maximal preidempotents of the idempotent e, and the summation goes over all nonempty subsets {i1,...,ik} of the set {1,...m}.
Abstract: In the semigroup algebra A of a finite inverse semigroup S over the field of complex numbers to an indempotent e there is assigned the sum σ(e)=e+∑(−1)KeL1⋯eiK, where ei,...,em are maximal preidempotents of the idempotent e, and the summation goes over all nonempty subsets {i1,...,ik} of the set {1,...m} Then for any class K of conjugate group elements of the semigroup S the element K=∑a·(a−1a) (the summation goes over all a∈g) is a central element of the algebra A, and the set {K} of all possible such elements is a basis for the center of the algebra A.
4 citations
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TL;DR: In this article, the canonical identification operator is characterized for Lax-Phillips evolutions, whose outgoing and incoming projections commute, and its spectral the-ory is considered in the special case, originally considered by Lax and Phillips, where the incoming and outgoing subspaces are mutually orthogonal.
Abstract: Lax-Phillipsevolutionsaredescribedbytwo-spacescatteringsys- tems. The canonical identification operator is characterized for Lax-Phillips evolutions, whose outgoing and incoming projections commute. In this case a (generalized) Lax-Phillips semigroup can be introduced and its spectral the- ory is considered. In the special case, originally considered by Lax and Phillips (where the outgoing and incoming subspaces are mutually orthogonal), this semigroup coincides with that introduced by Lax and Phillips. The basic con- nection of the Lax-Phillips semigroup to the so-called characteristic semigroup of the reference evolution is emphasized.
4 citations
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TL;DR: The aim of this note is to describe RD GCn-commutative permutable semigroups and RDGCn- Commutative ∆-semigroups, a generalization of generalized conditionally commutative semig groups.
Abstract: In this paper, we describe the RDGCn-commutative permutable semigroups and RDGCn-commutative ∆-semigroups. A semigroup S is called permutable, if for each pair of congruences ρ, σ on S, ρ ◦ σ = σ ◦ ρ. S is said to be a ∆-semigroup if the lattice of all congruences of S is a chain with respect to inclusion. Nagy, Schein, Tamura and Trotter etc. investigated the ∆-semigroups in special classes of semigroups([5], [6], [9]–[12]) and Cherubini, Varisco, Hamilton and the first author etc. investigated the permutable semigroups in special classes of semigroups ([1], [3], [4]). The aim of this note is to describe RDGCn-commutative permutable semigroups and RDGCn-commutative ∆-semigroups. A semigroup S is said to be right duo if, every right ideal of S is a two side ideal. We say that S is generalized conditionally commutative if S satisfies the identity xyx = xyx for every positive integer i ([8]). As a generalization of generalized conditionally commutative semigroups, Nagy [6] introduced the following definition: Definition. For a positive integer n, a semigroup will be called generalized conditionally n-commutative (or GCn-commutative) if it satisfies the identity xyx = xyx for every positive integer i ≥ 2. A semigroup is said to be RDGCn − commutative if it is both right duo and GCn-commutative. Clearly, a band is a GCn-commutative semigroup for every positive integer n. It is easy to see that a group is GCn-commutative if and only if it is commutative, and that GC1-commutative semigroups are just generalized conditionally commutative semigroups. Mathematics subject classification number: 20M10.
4 citations