Narrow Results

HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base, are defined by means of a construction based upon the isomorphism between the categories of inverse semigroups and inductive groupoids. The resulting HNN extension may conveniently be described by an inverse semigroup presentation, and we determine when an HNN extension with finitely generated or finitely presented base is again finitely generated or finitely presented. Our main results depend upon properties of the -preorder in the associated subsemigroups. Let S be a finitely generated inverse semigroup and let U, V be inverse subsemigroups of S, isomorphic via φ: U → V, that are order ideals in S. We prove that the HNN extension S*_U,φ is finitely generated if and only if U is finitely -dominated. If S is finitely presented, we give a necessary and suffcient condition for S*_U,φ to be finitely presented. Here, in contrast to the theory of HNN extensions of groups, it is not necessary that U be finitely generated.

There is a well-known correspondence between varieties of algebras and fully invariant congruences on the appropriate term algebra. A special class of varieties are those which are balanced, meaning they can be described by equations in which the same variables appear on each side. In this paper, we prove that the above correspondence, restricted to balanced varieties, leads to a correspondence between balanced varieties and inverse monoids. In the case of unary algebras, we recover the theorem of Meakin and Sapir that establishes a bijection between congruences on the free monoid with n generators and wide, positively self-conjugate inverse submonoids of the polycyclic monoid on n generators.

In the case of varieties generated by linear equations, meaning those equations where each variable occurs exactly once on each side, we can replace the clause monoid above by the linear clause monoid. In the case of algebras with a single operation of arity n, we prove that the linear clause monoid is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated essential right ideals of the free monoid on n letters, a monoid previously studied by Birget in the course of work on the Thompson group V and its analogues.

We show that Dehornoy's geometry monoid of a balanced variety is a special kind of inverse submonoid of ours.

Finally, we construct groups from the inverse monoids associated with a balanced variety and examine some conditions under which they still reflect the structure of the underlying variety. Both free groups and Thompson's groups V_n,1 arise in this way.

We show how one can use the partialization functor to obtain several recently defined inverse monoids, and use this functor to define new objects, which we call the inverse braid-permutation monoids. A presentation for this monoid is obtained. Finally, we study some abstract properties of the partialization functor and its iterations. This leads to a categorification of a monoid of all order-preserving maps, and series of orthodox generalizations of the symmetric inverse semigroup.

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S₁, S₂; U] of two finite inverse semigroups S₁, S₂ whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max{|S₁|,|S₂|}. Moreover we consider amalgams of finite inverse semigroups respecting the -order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the -classes to be finite.

The inverse semigroup of partial automaton permutations over a finite alphabet is characterized in terms of wreath products. The permutation conjugacy relation in this semigroup and the Green's relations are described. Criteria of primary conjugacy and conjugacy are given for certain naturally defined families of partial automaton permutations. Sufficient conditions under which an inverse semigroup admits a level transitive action are presented. We give explicit examples (monogenic inverse semigroups and some commutative Clifford semigroups) of inverse semigroups generated by finite automata.

We generalize the word problem for groups, the formal language of all words in the generators that represent the identity, to inverse monoids. In particular, we introduce the idempotent problem, the formal language of all words representing idempotents, and investigate how the properties of an inverse monoid are related to the formal language properties of its idempotent problem. We show that if an inverse monoid is either E-unitary or has a finite set of idempotents, then its idempotent problem is regular if and only if the inverse monoid is finite. We also give examples of inverse monoids with context-free idempotent problems, including all Bruck–Reilly extensions of finite groups.

We introduce the concept of an extension of a semilattice of groups $A$ by a group $G$ and describe all the extensions of this type which are equivalent to the crossed products $A{\ast }_{\mathrm{\Theta }}G$ by twisted partial actions $\mathrm{\Theta }$ of $G$ on $A$. As a consequence, we establish a one-to-one correspondence, up to an isomorphism, between twisted partial actions of groups on semilattices of groups and so-called Sieben twisted modules over $E$-unitary inverse semigroups.

We give necessary and sufficient conditions in order that lower bounded HNN-extensions of inverse semigroups and HNN-extensions of finite inverse semigroups are completely semisimple semigroups. Since it is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup, we first characterize such HNN-extensions containing a bicyclic subsemigroup making use of the special feature of their Schützenberger automata.

In this paper, we provide an abstract characterization of the inverse hulls of semigroups associated with Markov shifts. As an application of the characterization we give an example of Markov shifts that are not conjugate, but have isomorphic inverse hulls.

Locally inverse semigroups are regular semigroups whose idempotents form pseudo-semilattices. We characterize the categories that correspond to locally inverse semigroups in the realm of Nambooripad’s cross-connection theory. Further, we specialize our cross-connection description of locally inverse semigroups to inverse semigroups and completely $0$-simple semigroups, obtaining structure theorems for these classes. In particular, we show that the structure theorem for inverse semigroups can be obtained using only one category, quite analogous to the Ehresmann–Schein–Nambooripad Theorem; for completely $0$-simple semigroups, we show that cross-connections coincide with structure matrices, thus recovering the Rees Theorem by categorical tools.

From any directed graph E one can construct the graph inverse semigroup $G(E)$, whose elements, roughly speaking, correspond to paths in E. Wang and Luo showed that the congruence lattice $L(G(E))$ of $G(E)$ is upper-semimodular for every graph E, but can fail to be lower-semimodular for some E. We provide a simple characterization of the graphs E for which $L(G(E))$ is lower-semimodular. We also describe those E such that $L(G(E))$ is atomistic, and characterize the minimal generating sets for $L(G(E))$ when E is finite and simple.

We investigate the relationship between inverse semigroups and a special class of seminear-rings. We show that when an inverse semigroup T generates a d.g. seminear-ring (S,T) then the representation of the seminear-ring (S,T) is faithful. Some other aspects of representation theory of seminear-rings are discussed in connection with the structure of the underlying semigroups.

In this paper, we introduce the notion of a partial action of an ordered groupoid on a ring and we construct the corresponding partial skew groupoid ring. We present sufficient conditions under which the partial skew groupoid ring is either associative or unital. Also, we show that there is a one-to-one correspondence between partial actions of an ordered groupoid G on a ring R, in which the domain of each partial bijection is an ideal, and meet-preserving global actions of the Birget–Rhodes expansion G^BR of G on R. Using this correspondence, we prove that the partial skew groupoid ring is a homomorphic image of the skew groupoid ring constructed through the Birget–Rhodes expansion.

We use a new description of Kachel semigroup as triples with a pushout product. A simple Clifford category as a canonical representative (under the relation of equivalence of categories) of the standard Clifford category of this semigroup is considered.

For $n\in \mathbb{N}$, let $X_n=\{a_1,a_2,\dots ,a_n\}$ be an $n$-element set and let $F=(X_n;{<}_f)$ be a fence, also called a zigzag poset. As usual, we denote by $I_n$ the symmetric inverse semigroup on $X_n$. We say that a transformation $\alpha \in I_n$ is fence-preserving if $x{<}_{f}y$ implies that $x\alpha {<}_{f}y\alpha$, for all $x,y$ in the domain of $\alpha$. In this paper, we study the semigroup $PF{I}_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $I{F}_n=\{\alpha \in PF{I}_n:{\alpha }^{-1}\in PF{I}_n\}$. Clearly, $I{F}_n$ is an inverse semigroup and contains all regular elements of $PF{I}_n.$ We characterize the Green’s relations for the semigroup $I{F}_n$. Further, we prove that the semigroup $I{F}_n$ is generated by its elements with rank $\ge n-2$. Moreover, for $n\in 2\mathbb{N}$, we find the least generating set and calculate the rank of $I{F}_n$.

Let $(S,+,\cdot )$ be a semiring generated by one element. Let us denote this element by $w\in S$ and let $g(x)\in x\cdot \mathbb{N}[x]$ be a polynomial. It has been proved that if $g(x)$ contains at least two different monomials, then the elements of the form $g(w)$ may possibly be contained in any countable commutative semigroup. In particular, divisibility of such elements does not imply their torsion. Let, on the other hand, $g(x)$ consist of a single monomial (i.e. $g(x)=k{x}^n$, where $k,n\in \mathbb{N}$). We show that in this case, the divisibility of $g(w)$ by infinitely many primes implies that $g(w)$ generates a group within $(S,+)$. Further, an element $a\in S$ is called an $m$-fraction of an element $z\in S$ if $m\in \mathbb{N}$ and $z=m\cdot a$. We prove that “almost every” $m$-fraction of $w^n$ can be expressed as $f(w)$ for some polynomial $f\in x\cdot \mathbb{N}[x]$ of degree at most $n$.

Links between Möbius functions and inverse semigroups are an interesting topic for study. In this paper, we restrict our interest to the submonoid of right units (as a Möbius monoid) of a combinatorial bisimple inverse monoid. The Möbius functions at the end of this paper are Möbius functions of broken Möbius categories (broken up into two parts) via a submonoid and a right monoid action.

We review various constructions of wreath products of groups, semigroups, Lie algebras and associative algebras and discuss their realizations in matrix wreath products of associative algebras. As an application we prove a new version of Evans’s embedding theorem [T. Evans, Embedding theorems for multiplicative systems and projective geometries, Proc. American Math. Soc. 3 (1952) 614–620].

We investigate the connections between amalgams and Yamamura's HNN-extensions of inverse semigroups. In particular, we prove that amalgams of inverse semigroups with an identity adjoint are quotient semigroups of some special Yamamura's HNN-extensions. As a consequence, we show how to obtain the Schützenberger graph of a word w with respect to the presentation of an amalgamated free product starting from the Schützenberger graph of a suitable word w′ with respect to the presentation of the associated HNN-extension by simply making V-quotients and deletions of edges.

We generalise the Margolis-Meakin graph expansion of a group to a construction for ordered groupoids, and show that the graph expansion of an ordered groupoid enjoys structural properties analogous to those for graph expansions of groups. We also use the Cayley graph of an ordered groupoid to prove a version of McAlister's P-theorem for incompressible ordered groupoids.