Semitransitive subsemigroups of the symmetric inverse semigroups

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.

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.

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.

There are two natural analogues of the symmetric group on n symbols in the theory of semigroups, namely, the set of all mappings of a set of n symbols into itself, and the set of all partial transformations of such a set, with the obvious definitions of multiplication. We are concerned here with the latter system. This is an inverse semigroup, and accordingly we call it the ‘symmetric inverse semigroup’. It gives rise to a semisimple algebra over a field of characteristic zero or a prime greater than n, and its matrix representations over such a field are thus completely reducible.

The representation [5] of the centralizerC(x) of a permutationx in (a symmetric inverse semigroup)C n involves direct products of wreath products. Indeed, this semigroup case extends its group theory counterpart. Here, the last case (forx nilpotent) is addressed: A quotient of a wreath product is introduced and used to obtain a representation of the correspondingC(x). It follows that, for anyx∈C n ,C(x) can be imbedded in a direct product of wreath products with a quotient of a wreath product. A formula for calculating the order ofC(x) is given. The independent parameters in the formula are precisely those that define the path structure ofx∈C n .

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.

Let ℐn be the symmetric inverse semigroup on Xn = {1, 2,…, n}, and let 𝒟𝒫n and 𝒪𝒟𝒫n be its subsemigroups of partial isometries and of order-preserving partial isometries of Xn, respectively. In this article, we investigate the cycle structure of a partial isometry and characterize the Green's relations on 𝒟𝒫n and 𝒪𝒟𝒫n. We show that 𝒪𝒟𝒫n is a 0-E-unitary inverse semigroup. We also investigate the ranks of 𝒟𝒫n and 𝒪𝒟𝒫n.

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].

Varieties generated by unstable involution semigroups with continuum many subvarieties

Let $ \mathcal{I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2, \ldots , n\}$. Let $\mathcal{OCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all order-preserving injective partial contraction mappings, and let $\mathcal{ODCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all order-preserving and order-decreasing injective partial contraction mappings of $X_n$. In this paper, we investigate the cardinalities of some equivalences on $\mathcal{OCI}_n$ and $\mathcal{ODCI}_n$ which lead naturally to obtaining the order of these semigroups. Then, we relate the formulae obtained to Fibonacci numbers. Similar results about $\mathcal{ORCI}_n$, the semigroup of order-preserving or order-reversing injective partial contraction mappings, are deduced.

We show how several famous combinatorial sequences appear in the context of nilpotent elements of the full symmetric inverse semigroup \( I \mathcal{S}_n \). These sequences appear either as cardinalities of certain nilpotent subsemigroups or as the numbers of special nilpotent elements and include the Lah numbers, the Bell numbers, the Stirling numbers of the second kind, the binomial coefficients and the Catalan numbers.

In 2003, Marques-Smith and Sullivan described the join Ω of thenatural order' and thecontainment order' on P ( X), the semigroup under composition of all partial transformations of a set X. And, in 2004, Pinto and Sullivan described all automorphisms of PS( q), the partial Baer-Levi semigroup consisting of all injective 2 P ( X) such that jX n X j = q, where @0 q j Xj. In this paper, we describe the group of automorphisms of R( q), the largest regular subsemigroup of PS( q). In 2010, we studied some properties of and on PS( q). Here, we characterize the meet and join under those orders for elements of R( q) and PS( q). In addition, since does not equal Ω on I( X), the symmetric inverse semigroup on X, we formulate an algebraic version of

Area law violations for entanglement entropy in the form of a square root have recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their variations. Here we consider a Motzkin walk with a different Hilbert space on each step of the walk spanned by the elements of a symmetric inverse semigroup with the direction of each step governed by its algebraic structure. This change alters the number of paths allowed in the Motzkin walk and introduces a ground state degeneracy that is sensitive to boundary perturbations. We study the frustration-free spin chains based on three symmetric inverse semigroups, , and . The system based on and provides examples of quantum phase transitions in one dimension, with the former exhibiting a transition between the area law and a logarithmic violation of the area law and the latter providing an example of transition from logarithmic scaling to a square root scaling in the system size, mimicking a colored system. The system with is much simpler and produces states that continue to obey the area law.

On F-inverse covers of inverse monoids

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.