Semigroup Forum最新文献

The class of identical inclusions was defined by E.S. Lyapin.This is the class of universal formulas which is situated strictly between identities and universal positive formulas.These universal formulas can be written as identical equalities of subsets of (X^+). Classes of semigroups defined by identical inclusions are called inclusive varieties. We describe finite inclusive varieties of semigroups and study countable inclusive varieties of semigroups.We also describe small inclusive varieties, that is, inclusive varieties with finite lattices of their inclusive subvarieties, of completely regular semigroups and study inclusive varieties of completely regular semigroups with countable lattices of their inclusive subvarieties

The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur–Weyl duality between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra. We also explain how this relates to Solomon’s Schur–Weyl duality between the rook monoid and the general linear group and mention some advantages of our approach.

In this paper we provide an account of the Todd–Coxeter algorithm for computing congruences on semigroups and monoids. We also give a novel description of an analogue for semigroups of the so-called Felsch strategy from the Todd–Coxeter algorithm for groups.

We give an account on what is known on the subject of permutation matchings, which are bijections of a finite regular semigroup that map each element to one of its inverses. This includes partial solutions to some open questions, including a related novel combinatorial problem.

We expand the concept of local embeddability into finite structures (LEF) for the class of semigroups with investigations of non-LEF structures, a closely related generalising property of local wrapping of finite structures (LWF) and inverse semigroups. The established results include a description of a family of non-LEF semigroups unifying the bicyclic monoid and Baumslag–Solitar groups and demonstrating that inverse LWF semigroups with finite number of idempotents are LEF.

From any poset isomorphic to the poset of gaps of a numerical semigroup S with the order induced by S, one can recover S. As an application, we prove that two different numerical semigroups cannot have isomorphic posets (with respect to set inclusion) of ideals whose minimum is zero. We also show that given two numerical semigroups S and T, if their ideal class monoids are isomorphic, then S must be equal to T.

Two semigroups are lattice isomorphic if their subsemigroup lattices are isomorphic, and a class of semigroups is lattice closed if it contains every semigroup which is lattice isomorphic to some semigroup from that class. An orthodox semigroup is a regular semigroup whose idempotents form a subsemigroup. We prove that the class of all orthodox semigroups having no nontrivial finite subgroups is lattice closed.

We give a pointwise version of sensitivity in terms of open covers for a semiflow (T, X) of a topological semigroup T on a Hausdorff space X and call it a Hausdorff sensitive point. If ((X, {mathscr {U}})) is a uniform space with topology (tau ), then the definition of Hausdorff sensitivity for ((T, (X, tau ))) gives a pointwise version of sensitivity in terms of uniformity and we call it a uniformly sensitive point. For a semiflow (T, X) on a compact Hausdorff space X, these notions (i.e. Hausdorff sensitive point and uniformly sensitive point) are equal and they are T-invariant if T is a C-semigroup. They are not preserved by factor maps and subsystems, but behave slightly better with respect to lifting. We give the definition of a topologically equicontinuous pair for a semiflow (T, X) on a topological space X and show that if (T, X) is a topologically equicontinuous pair in (x, y), for all (yin X), then (overline{Tx}= D_T(x)) where

We prove for a topologically transitive semiflow (T, X) of a C-semigroup T on a regular space X with a topologically equicontinuous point that the set of topologically equicontinuous points coincides with the set of transitive points. This implies that every minimal semiflow of C-semigroup T on a regular space X with a topologically equicontinuous point is topologically equicontinuous. Moreover, we show that if X is a regular space and (T, X) is not a topologically equicontinuous pair in (x, y), then x is a Hausdorff sensitive point for (T, X). Hence, a minimal semiflow of a C-semigroup T on a regular space X is either topologically equicontinuous or topologically sensitive.

Consider the set (M_{a,b} = {n in mathbb {Z}_{ge 1}: n equiv a bmod b} cup {1}) for (a, b in mathbb {Z}_{ge 1}). If (a^2 equiv a bmod b), then (M_{a,b}) is closed under multiplication and known as an arithmetic congruence monoid (ACM). A non-unit (n in M_{a,b}) is an atom if it cannot be expressed as a product of non-units, and the atomic density of (M_{a,b}) is the limiting proportion of elements that are atoms. In this paper, we characterize the atomic density of (M_{a,b}) in terms of a and b.

We observe that Beck modules for a commutative monoid are exactly modules over a graded commutative ring associated to the monoid. Under this identification, the Quillen cohomology of commutative monoids is a special case of the André–Quillen cohomology for graded commutative rings, generalizing a result of Kurdiani and Pirashvili. To verify this we develop the necessary grading formalism. The partial cochain complex developed by Pierre Grillet for computing Quillen cohomology appears as the start of a modification of the Harrison cochain complex suggested by Michael Barr.