Ranks and presentations of some normally ordered inverse semigroups

It is shown that every finite inverse semigroup having only solvable subgroups,viewed as a semigroup with the additional unary operation of inversion, has nofinite basis of identities, unless it is a strict inverse semigroup.

Given any quasi-countable, in particular, any countable inverse semigroup S , we introduce a way to equip S with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on S . This allows us to unambiguously define the uniform Roe algebra of S , which we prove can be realized as a canonical crossed product of \ell^{\infty}(S) and S . We relate these metrics to the analogous metrics on Hausdorff étale groupoids. Using this setting, we study those inverse semigroups with asymptotic dimension 0 . Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by S being locally \mathcal{L} -finite, and equivalently sparse as a metric space.

The minimal degree of an inverse semigroup S is the minimal cardinality of a set A such that S is isomorphic to an inverse semigroup of one-to-one partial transformations of A .The main result is a formula that expresses the minimal degree of a finite inverse semigroup S in terms of certain subgroups and the ordered structure of S .In fact, a representation of S by one-to-one partial transformations of the smallest possible set A is explicitly constructed in the proof of the formula.All known and some new results on the minimal degree follow as easy corollaries.

For [Formula: see text], let [Formula: see text] be an [Formula: see text]-element set and let [Formula: see text] be a fence, also called a zigzag poset. As usual, we denote by [Formula: see text] the symmetric inverse semigroup on [Formula: see text]. We say that a transformation [Formula: see text] is fence-preserving if [Formula: see text] implies that [Formula: see text], for all [Formula: see text] in the domain of [Formula: see text]. In this paper, we study the semigroup [Formula: see text] of all partial fence-preserving injections of [Formula: see text] and its subsemigroup [Formula: see text]. Clearly, [Formula: see text] is an inverse semigroup and contains all regular elements of [Formula: see text] We characterize the Green’s relations for the semigroup [Formula: see text]. Further, we prove that the semigroup [Formula: see text] is generated by its elements with rank [Formula: see text]. Moreover, for [Formula: see text], we find the least generating set and calculate the rank of [Formula: see text].

We show that every such semigroup is a homomorphic image of a subsemigroup of some finite inverse semigroup. This shows that the pseudovariety generated by the finite inverse semigroups consists of exactly the finite semigroups with commuting idempotents.

On F-inverse covers of inverse monoids

In semigroup theory as in other algebraic theories a significant part of the total effort is appropriately applied to the study of certain standard examples occurring, as it were, “in nature”. The most obvious such semigroup is the full transformation semigroup (X) (see [3]) and about this semigroup a great deal is known in both the finite and infinite cases.

We initiate the study of semitransitive transformation semigroups. In the paper we describe the structure of semitransitive subsemigroups of the finite symmetric inverse semigroup of the minimal cardinality modulo the classification of transitive subgroups of the minimal cardinality of finite symmetric groups, and state the results on minimal transitive subsemigroups.

The commuting graph of a finite non-commutative semigroup $S$, denoted $\cg(S)$, is a simple graph whose vertices are the non-central elements of $S$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Let $\mi(X)$ be the symmetric inverse semigroup of partial injective transformations on a finite set $X$. The semigroup $\mi(X)$ has the symmetric group $\sym(X)$ of permutations on $X$ as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of $\sym(X)$. In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of $\cg(\sym(X))$, and in 2011, Dol\u{z}an and Oblak claimed (but their proof has a GAP) that this upper bound is in fact the exact value. The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup $\mi(X)$. We calculate the clique number of $\cg(\mi(X))$, the diameters of the commuting graphs of the proper ideals of $\mi(X)$, and the diameter of $\cg(\mi(X))$ when $|X|$ is even or a power of an odd prime. We show that when $|X|$ is odd and divisible by at least two primes, then the diameter of $\cg(\mi(X))$ is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of $\mi(X)$ of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of $\mi(X)$. The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.

Let G be an arbitrary group of bijections on a finite set and let I(G) denote the set of all partial injective transformations each of which is included in a bijection from G. The set I(G) is a fundamental factorizable inverse semigroup. We study various properties of the semigroup I(G). In particular, we describe the automorphisms of I(G) and obtain necessary and sufficient conditions for each stable order on I(G) to be fundamental or antifundamental.

A natural class of sentences about the lattice of recursively enumerable sets modulo finite sets is shown to be decidable.This class properly contains the class of sentences previously shown to be decidable by Lachlan.New structure results about the lattice of recursively enumerable sets are proved which play an important role in the decision procedure.0. Introduction.Much of the recent work dealing with S, the lattice of recursively enumerable sets, has dealt with global properties of S such as automorphisms and decidability, rather than local properties of S, i.e., properties of definable classes of recursively enumerable sets.Two of the major results are Lachlan's [3] decision procedure for a natural fragment of the elementary theory of S *, the quotient lattice of S by the ideal of finite sets, Soare's result [15] on the existence of automorphisms carrying any maximal set into any other maximal set.More recently, Shore [13] has determined the definable automorphism bases for .These global results have inspired new local results, in that they have naturally led to the discovery of important new S-definable classes of recursively enumerable sets whose properties have been investigated.Lachlan's result led to the discovery of small sets, Soare's result led to the discovery of /-simple sets, and Shore's result led to the discovery of nowhere simple sets; the first two of these classes have been studied by Lerman and Soare [8], and the third class by Shore [14].The class of -simple sets proved to be of particular importance, in that it led to the refutation of conjectures of Martin and Shoenfield which imply that the degrees of elements of any S-definable class can be characterized by a finite set of equalities and inequalities involving the jumps of those degrees.Evidence for these conjectures included Martin's result [9] that a is the degree of a maximal set if and only if a' = 0", and results of Lachlan [2] and Shoenfield [12] that a is the degree of an atomless set if and only if a" > 0".In this paper, we give a decision procedure for a larger fragment of the

A fence is a particular partial order on a (finite) set, close to the linear order. In this paper, we calculate the rank of the semigroup $$\mathscr {FI}_{n}$$ of all order-preserving partial injections on an n-element fence. In particular, we provide a minimal generating set for $$\mathscr {FI}_{n}$$ . In the present paper, n is odd since this problem for even n was already solved by I. Dimitrova and J. Koppitz.

On representing Mn's by congruence lattice of finite algebras

We prove that the word problem is decidable in Yamamura’s HNN extensions of finite inverse semigroups, by providing an iterative construction of approximate automata of the Schutzenberger automata of words relative to the standard presentation of Yamamura’s HNN-extensions.

We use the description of the Schutzenberger automata for amalgams of finite inverse semigroups given by Cherubini, Meakin, Piochi to obtain structural results for such amalgams. Schutzenberger automata, in the case of amalgams of finite inverse semigroups, are automata with special structure possessing finite subgraphs, that contain all essential information related to the whole automaton. Using this crucial fact, and the Bass-Serre theory, we show that the maximal subgroups of an amalgamated free-product are either isomorphic to certain subgroups of the original semigroups or can be described as fundamental groups of particular finite graphs of groups build from the maximal subgroups of the original semigroups.