On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers

In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

Maximal inverse subsemigroups of the semigroup of all full transformations satisfying x ≈ x3

We classify the maximal Clifford inverse subsemigroups [Formula: see text] of the full transformation semigroup [Formula: see text] on an [Formula: see text]-element set with [Formula: see text] for all [Formula: see text]. This classification differs from the already known classifications of Clifford inverse semigroups, it provides an algorithm for its construction. For a given natural number [Formula: see text], we find also the largest size of an inverse subsemigroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] with least rank [Formula: see text] for any element in [Formula: see text].

Combinatorially Factorizable Cryptic Inverse Semigroups

An inverse semigroup S is combinatorially factorizable if S = TG where T is a combinatorial (i.e., 𝓗 is the equality relation) inverse subsemigroup of S and G is a subgroup of S. This concept was introduced and studied byMills, especially in the case when S is cryptic (i.e., 𝓗 is a congruence on S). Her approach is mainly analytical considering subsemigroups of a cryptic inverse semigroup.We start with a combinatorial inverse monoid and a factorizable Clifford monoid and from an action of the former on the latter construct the semigroups in the title. As a special case, we consider semigroups that are direct products of a combinatorial inverse monoid and a group.

Ideal extensions of locally inverse semigroups

The translational hull of a locally inverse semigroup has a largest locally inverse subsemigroup containing the inner part. A construction is given for ideal extensions within the class of all locally inverse semigroups.

EMBEDDING ℐn IN A 2-GENERATOR INVERSE SUBSEMIGROUP OF ℐn+2

Given an integer $n$, we show that $\mathcal{I}_{n}$ embeds in a 2-generated subsemigroup of $\mathcal{I}_{n+2}$, which is an inverse semigroup. An immediate consequence of this result is the following, which is analogous to the case for groups and semigroups: every finite inverse semigroup may be embedded in a finite 2-generated semigroup which is an inverse semigroup.AMS 2000 Mathematics subject classification: Primary 20M18. Secondary 20M20

On quasi-orthodox semigroups with inverse transversals

An inverse transversal of a regular semigroup S is an inverse subsemigroup So that contains precisely one inverse of each element of S. Here we consider the case where S is quasi-orthodox. We give natural characterisations of such semigroups and consider various properties of congruences.

A simplistic approach to inverse transversals

An inverse transversal of a regular semigroup S is an inverse subsemigroup that contains precisely one inverse of each element of S. In the literature there are three known types of inverse transversal, namely those that are multiplicative, those that are weakly multiplicative, and those that form quasi-ideals. Here, by considering natural ways in which certain words can be simplified, we reveal four new types of inverse transversal. All of these can be illustrated nicely in examples that are based on 2 × 2 matrices.