Endomorphisms of Cayley digraphs of rectangular groups

Let Cay(S,A) denote the Cayley digraph of the semigroup S with respect to the set A, where A is any subset of S. The function f : Cay(S,A) → Cay(S,A) is called an endomorphism of Cay(S,A) if for each (x, y) ∈ E(Cay(S,A)) implies (f(x), f(y)) ∈ E(Cay(S,A)) as well, where E(Cay(S,A)) is an arc set of Cay(S,A). We characterize the endomorphisms of Cayley digraphs of rectangular groups G × L × R, where the connection sets are in the form of A = K × P × T

Certain structural properties for Cayley regularity graphs of semigroups and their theoretical applications

An element $x$ in a semigroup is said to be regular if there exists an element $y$ in the semigroup such that $x = xyx$. The element $y$ is said to be a regular part of $x$. Define the Cayley regularity graph of a semigroup $S$ to be a digraph with vertex set $S$ and arc set containing all ordered pairs $(x, y)$ such that $y$ is a regular part of $x$. In this paper, certain classes of Cayley regularity graphs such as complete digraphs, connected digraphs and equivalence digraphs are investigated. Furthermore, structural properties of the Cayley regularity graphs are theoretically applied to study perfect matchings of other algebraic graphs

Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory

In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and \v{C}ern\'y's conjecture for an important class of automata

On endomorphism universality of sparse graph classes

Solving a problem of Babai and Pultr from 1980 we show that every commutative idempotent monoid (a.k.a lattice) is the endomorphism monoid of a graph of bounded degree. Indeed we show that maximum degree $3$ suffices, which is best-possible. On the way we generalize a classic result of Frucht by showing that every group is the endomorphism monoid of a graph of maximum degree $3$ and we answer a question of Ne\v{s}et\v{r}il and Ossona de Mendez from 2012, presenting a class of bounded expansion such that every monoid is the endomorphism monoid of one of its members. On the other hand we strengthen a result of Babai and Pultr and show that no class excluding a topological minor can have all completely regular monoids among its endomorphism monoids. Moreover, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids.Comment: 37 pages, 18 figure

The biHecke monoid of a finite Coxeter group and its representations

For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H_0(W), that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each w in W a combinatorial module T_w whose support is the interval [1,w]_R in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.Comment: v2: Added complete description of the rank 2 case (Section 7.3) and improved proof of Proposition 7.5. v3: Final version (typo fixes, picture improvements) 66 pages, 9 figures Algebra and Number Theory, 2013. arXiv admin note: text overlap with arXiv:1108.4379 by other author

Representation theory of finite semigroups, semigroup radicals and formal language theory

In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; andČerný's conjecture for an important class of automata

Classical Set Theory: Theory of Sets and Classes

This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-graduate) student in Mathematics should know about foundations of this science.Comment: 162 page