Inverse monoids and rational Schreier subsets of the free group

An inverse monoidM is an idempotent-pure image of the free inverse monoid on a setX if and only ifM has a presentation of the formM=Inv<X:e_o=f_i, i∈I>, wheree _i ,f _i are idempotents of the free inverse monoid: every inverse monoid is an idempotent-separating image of one of this type. IfR is anR-class of such an inverse monoid, thenR may be regarded as a Schreier subset of the free group onX. This paper is concerned with an examination of which Schreier subsets arise in this way. In particular, ifI is finite, thenR is a rational Schreier subset of the free group. Not every rational Schreier set arises in this way, but every positively labeled rational Schreier set does. Research supported by National Science Foundation grant #DMS8702019.     

Inverse monoids of graphs

. IntroductionGraph endomorphism and its regularity property have been investigated in some literatures (of. [1--41 for examples). The invertibility is a stronger algebraic property thanregUlarity in semigroup theory. It is commonly agreed that inverse semigroups are the mostpromising class of semigroups for study. In this paper we first present a combinatorial characterization of an inverse monoid of a graph (Theorem 2.3). Then using this we prove thata bipartite graph with an inverse monoi…     

Sparse and slender subsets of monoids

We generalize the notions of sparse and slender sets for an arbitrary monoid and characterize the unambiguous rational sets which are sparse or slender.     

Inverse monoids of partial graph automorphisms

Journal of Algebraic Combinatorics - A partial automorphism of a finite graph is an isomorphism between its vertex-induced subgraphs. The set of all partial automorphisms of a given finite graph...     

Artin monoids inject in their groups

We prove that the natural homomorphism from an Artin monoid to its associated Artin group is always injective. Received: March 14, 2002     

Partial symmetry, reflection monoids and Coxeter groups

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.     

Small subsets of groups

Given an infinite group G and an infinite cardinal κ|G|, we say that a subset A of G is κ-large (κ-small) if there exists F[G]^<κ such that G=FA (GFA is κ-large for each F[G]^<κ). The subject of the paper is the family of all κ-small subsets. We describe the left ideal of the right topological semigroup βG determined by . We study interrelations between κ-small and other (P_κ-small and κ-thin) subsets of groups, and prove that G can be generated by some 2-thin subsets. We partition G in countable many subsets which are κ-small for each κω. We show that [G]^<κ is dual to provided that either κ is regular and κ=|G|, or G is Abelian and κ is a limit cardinal, or G is a divisible Abelian group.     

Coxeter groups,Coxeter monoids and the Bruhat order

Associated with any Coxeter group is a Coxeter monoid, which has the same elements, and the same identity, but a different multiplication. (Some authors call these Coxeter monoids 0-Hecke monoids, because of their relation to the 0-Hecke algebras—the q=0 case of the Hecke algebra of a Coxeter group.) A Coxeter group is defined as a group having a particular presentation, but a pair of isomorphic groups could be obtained via non-isomorphic presentations of this form. We show that when we have both the group and the monoid structure, we can reconstruct the presentation uniquely up to isomorphism and present a characterisation of those finite group and monoid structures that occur as a Coxeter group and its corresponding Coxeter monoid. The Coxeter monoid structure is related to this Bruhat order. More precisely, multiplication in the Coxeter monoid corresponds to element-wise multiplication of principal downsets in the Bruhat order. Using this property and our characterisation of Coxeter groups among structures with a group and monoid operation, we derive a classification of Coxeter groups among all groups admitting a partial order.     

Quasi-equational bases for graphs of semigroups,monoids and groups

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. Let denote by the class of all graphs of algebras from a class . We prove that if is a class of semigroups possessing a nontrivial member with a neutral element, then does not have finite quasi-equational basis. We deduce that, for a class of monoids or groups with a nontrivial member, also does not have finite quasi-equational basis.     

The endomorphism monoids and automorphism groups of Cayley graphs of semigroups

In this note, we introduce the notions of color-permutable automorphisms and color-permutable vertex-transitive Cayley graphs of semigroups. As a main result, for a finite monoid S and a generating set C of S, we explicitly determine the color-permutable automorphism group of \(\mathrm {Cay}(S,C)\) [Theorem 1.1]. Also for a monoid S and a generating set C of S, we explicitly determine the color-preserving automorphism group (endomorphism monoid) of \(\mathrm {Cay}(S,C)\) [Proposition 2.3 and Corollary 2.4]. Then we use these results to characterize when \(\mathrm {Cay}(S,C)\) is color-endomorphism vertex-transitive [Theorem 3.4].     

Decidability of first-order theories for groups and monoids of integral matrices

We deal with the decidability problem for first-order theories of a complete linear group GL(n,ℤ) of all integral matrices of order n ≥ 3. and of a respective complete linear monoid ML(n,ℤ). It is proved that theories ∀? ∧ GL(3,ℤ). ∃∀∧ GL(3,ℤ). ∀? ∧ ML(3,ℤ), and ∃? ∧ ML(3,ℤ) are critical. and that ∃∀ ∧ νGL(n,ℤ) and ∃∀ ∧ML(n,ℤ) are decidable for any n ≥ 3. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 480–504, July–August, 2000.     

Representations of monoids arising from finite groups of Lie type

A class of finite monoids constructed from a group of Lie type is considered. We describe the irreducible complex representations and prove the complete reducibility of the representations of . The sandwich matrix of is decomposed into a product of matrices corresponding to maximal parabolic subgroups of .

     

Relatively thin and sparse subsets of groups

Let G be a group with identity e and let I \mathcal{I} be a left-invariant ideal in the Boolean algebra P_G {\mathcal{P}_G} of all subsets of G. A subset A of G is called I \mathcal{I} -thin if gA ?A ? I gA \cap A \in \mathcal{I} for every g ∈ G\{e}. A subset A of G is called I \mathcal{I} -sparse if, for T every infinite subset S of G, there exists a finite subset F ⊂ S such that ?_{g ? F} gA ? F \bigcap\nolimits_{g \in F} {gA \in \mathcal{F}} . An ideal I \mathcal{I} is said to be thin-complete (sparse-complete) if every I \mathcal{I} -thin (I \mathcal{I} -sparse) subset of G belongs to I \mathcal{I} . We define and describe the thin-completion and the sparse-completion of an ideal in P_G {\mathcal{P}_G} .     

Sets of minimal distances and characterizations of class groups of Krull monoids

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then every non-unit \(a \in H\) can be written as a finite product of atoms, say \(a=u_1 \cdot \ldots \cdot u_k\). The set \(\mathsf L (a)\) of all possible factorization lengths k is called the set of lengths of a. There is a constant \(M \in \mathbb N\) such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference \(d \in \Delta ^* (H)\), where \(\Delta ^* (H)\) denotes the set of minimal distances of H. We study the structure of \(\Delta ^* (H)\) and establish a characterization when \(\Delta ^*(H)\) is an interval. The system \(\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}\) of all sets of lengths depends only on the class group G, and a standing conjecture states that conversely the system \(\mathcal L (H)\) is characteristic for the class group. We confirm this conjecture (among others) if the class group is isomorphic to \(C_n^r\) with \(r,n \in \mathbb N\) and \(\Delta ^*(H)\) is not an interval.