Clifford Monoid Research Articles

Inner automorphisms of Clifford monoids

An automorphism $\phi$ of a monoid $S$ is called inner if there exists $g$ in $U_{S}$, the group of units of $S$, such that $\phi(s)=gsg^{-1}$ for all $s $ in $S$; we call $S$ nearly complete if all of its automorphisms are inner. In this paper, first we prove several results on inner automorphisms of a general monoid and subsequently apply them to Clifford monoids. For certain subclasses of the class of Clifford monoids, we give necessary and sufficient conditions for a Clifford monoid to be nearly complete. These subclasses arise from conditions on the structure homomorphisms of the Clifford monoids: all being either bijective, surjective, injective, or image trivial.

SEPARABILITY CONDITIONS IN ACTS OVER MONOIDS

We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and strongly subact separable (and hence weakly subact separable); all acts over a commutative idempotent monoid are residually finite and strongly subact separable; and all acts over a Clifford monoid are strongly subact separable.

Weak Hopf Algebra and Its Quiver Representation

This study induced a weak Hopf algebra from the path coalgebra of a weak Hopf quiver. Moreover, it gave a quiver representation of the said algebra which gives rise to the various structures of the so-called weak Hopf algebra through the quiver. Furthermore, it also showed the canonical representation for each weak Hopf quiver. It was further observed that a Cayley digraph of a Clifford monoid can be embedded in its corresponding weak Hopf quiver of a Clifford monoid. This lead to the development of the foundation structures of weak Hopf algebra. Such quiver representation is useful for the classification of its path coalgebra. Additionally, some structures of module theory of algebra were also given. Such algebras can also be applied for obtaining the solutions of “quantum Yang–Baxter equation” that has many applications in the dynamical systems for finding interesting results.

On formations of monoids

A formation of monoids is a class of finite monoids closed under taking quotients and subdirect products. Formations of monoids were first studied in connection with formal language theory, but in this paper, we come back to an algebraic point of view. We give two natural constructions of formations based on constraints on the minimal ideal and on the maximal subgroups of a monoid. Next we describe two sublattices of the lattice of all formations, and give, for each of them, an isomorphism with a known lattice of varieties of monoids. Finally, we study formations and varieties containing only Clifford monoids, completely describe such varieties and discuss the case of formations.

Unit regular inverse monoids and Cliford monoids

Let S be a unit regular semigroup with group of units G = G(S) and semilattice of idempotents E = E(S). Then for every there is a such that Then both xu and ux are idempotents and we can write or .Thus every element of a unit regular inverse monoid is a product of a group element and an idempotent. It is evident that every L-class and every R-class contains exactly one idempotent where L and R are two of Greens relations. Since inverse monoids are R unipotent, every element of a unit regular inverse monoid can be written as s = eu where the idempotent part e is unique and u is a unit. A completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. A Clifford semigroup is a completely regular inverse semigroup. Characterization of unit regular inverse monoids in terms of the group of units and the semilattice of idempotents is a problem often attempted and in this direction we have studied the structure of unit regular inverse monoids and Clifford monoids.

Combinatorially Factorizable Cryptic Inverse Semigroups

AbstractAn 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.

Constructing pointed Hopf algebra by Ore-extensions

The main work of this article is to give a nontrivial method to construct pointed semilattice graded weak Hopf algebra ▪ from a Clifford monoid S = [Y;Gα, φα,β] by Ore-extensions, and to obtain a co-Frobenius semilattice graded weak Hopf algebra H(S, n, c, χ, a, b) through factoring At by a semilattice graded weak Hopf ideal.

On Some Finitary Conditions Arising from the Axiomatisability of Certain Classes of Monoid Acts

This article considers those monoids S satisfying one or both of the finitary properties (R) and (r), focussing for the most part on inverse monoids. These properties arise from questions of axiomatisability of classes of S-acts, and appear to be of interest in their own right. If S is weakly right noetherian (WRN), that is, S has the ascending chain condition on right ideals, then certainly (r) holds. Other than this, we show that (R), (r), and (WRN) are independent. Our most detailed results are for Clifford monoids, in which case we completely characterise those S with trivial structure homomorphisms satisfying (R) or (r).

Monoid-matrix type automata

Monoid-matrix type automata are introduced and studied in this paper. We give a characterization of the cyclic monoid-matrix type automata and the regular monoid-matrix type automata. Also, we provide a method to determine the structures of canonical Sℓ-automata (canonical C-automata, respectively) whose endomorphism monoids are isomorphic to a given finite meet semilattice with the greatest element (Clifford monoid, respectively).

On Endomorphisms of Ockham Algebras with Pseudocomplementation

A pO-algebra $${(L; f, \, ^{\star})}$$ is an algebra in which (L; f) is an Ockham algebra, $${(L; \, ^{\star})}$$ is a p-algebra, and the unary operations f and $${^{\star}}$$ commute. Here we consider the endomorphism monoid of such an algebra. If $${(L; f, \, ^{\star})}$$ is a subdirectly irreducible pK 1,1- algebra then every endomorphism $${\vartheta}$$ is a monomorphism or $${\vartheta^3 = \vartheta}$$ . When L is finite the endomorphism monoid of L is regular, and we determine precisely when it is a Clifford monoid.

Representations of commutative asynchronous automata

Ito (1976, 1978) [14,17] provided representations of strongly connected automata by group-matrix type automata. This shows the close connection between strongly connected automata with their automorphism groups. In this paper we deal with commutative asynchronous automata. In particular, we introduce and study normal commutative asynchronous automata and cyclic commutative asynchronous automata. Some properties on endomorphism monoids of these automata are given. Also, the representations of normal commutative asynchronous automata and cyclic commutative asynchronous automata are provided by S-automata and regular S-automata, respectively. The cartesian composition A ∘ B of a strongly connected automaton A and a cyclic commutative asynchronous automaton B is studied. It is shown that the endomorphism monoid E ( A ∘ B ) of automaton A ∘ B is a Clifford monoid. Finally, a representation of A ∘ B is provided by regular Clifford monoid matrix-type automaton. This generalizes and extends the representations of strongly connected automata given by Ito (1976) [14].

Homological finiteness properties of monoids, their ideals and maximal subgroups

We consider the general question of how the homological finiteness property left- FP n (resp. right- FP n ) holding in a monoid influences, and conversely depends on, the property holding in the substructures of that monoid. This is done by giving methods for constructing free resolutions of substructures from free resolutions of their containing monoids, and vice versa. In particular, we show that left- FP n is inherited by the maximal subgroups in a completely simple minimal ideal, in the case that the minimal ideal has finitely many left ideals. For completely simple semigroups, we prove the converse, and as a corollary, show that a completely simple semigroup is of type left- and right- FP n if and only if it has finitely many left and right ideals and all of its maximal subgroups are of type FP n . Also, given an ideal of a monoid, we show that if the ideal has a two-sided identity element then the containing monoid is of type left- FP n if and only if the ideal is of type left- FP n . Applying this result, we obtain necessary and sufficient conditions for a Clifford monoid (and more generally a strong semilattice of monoids) to be of type left- FP n . Examples are provided showing that for each of the results all of the hypotheses are necessary.

Weak Hopf Quivers, Clifford Monoids and Weak Hopf Algebras

The purpose of this paper is to use Clifford monoids to define weak Hopf quivers and to construct the corresponding class of weak Hopf algebras, i.e., the semilattice of graded weak Hopf algebras which were found to be useful in obtaining a regular R-matrix of a quantum double without a quasi-triangular structure. We define weak Hopf quivers and classify the semilattice of graded weak Hopf algebra structures over path coalgebras using weak Hopf quivers. This gives, precisely, a generalization of the classification of the graded weak Hopf algebra structures over path coalgebras using such quivers.

On Presentations of Bruck–Reilly Extensions

We first consider the class of monoids in which every left invertible element is also right invertible, and prove that if a monoid belonging to this class admits a finitely presented Bruck–Reilly extension then it is finitely generated. This allow us to obtain necessary and sufficient conditions for the Bruck–Reilly extensions of this class of monoids to be finitely presented. We then prove that thes 𝒟-classes of a Bruck–Reilly extension of a Clifford monoid are Bruck–Reilly extensions of groups. This yields another necessary and sufficient condition for these Bruck–Reilly extensions to be finitely generated and presented. Finally, we show that a Bruck–Reilly extension of a Clifford monoid is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid, and that this property cannot be generalized to Bruck–Reilly extensions of arbitrary inverse monoids.

Artin’s Equivalence Generalized

Let S be a residuated commutative monoid. Artin’s classical equivalence is related to a semigroup homomorphism of S onto the group of divisorial elements in case these form a group. An equivalence relation is considered that is more general than Artin’s equivalence. We introduce relatively divisorial elements and proceed to investigate conditions which ensure that S admits a homomorphism onto the semigroup of relatively divisorial elements when this semigroup is a Clifford monoid.

FINITE PRESENTABILITY OF BRUCK–REILLY EXTENSIONS OF CLIFFORD MONOIDS

Let M be a Clifford monoid and let θ be an endomorphism of M. We prove that if the Bruck–Reilly extension BR(M, θ) is finitely presented then M is finitely generated. This allows us to derive necessary and sufficient conditions for Bruck–Reilly extensions of Clifford monoids to be finitely presented.

Structure of Quantum Doubles of Finite Clifford Monoids#

In this paper, over a field k, we give the structure theorem of the quantum double of a finite Clifford monoid through bicrossed products and quantum doubles of groups. By this result, it is shown that the quantum double of a finite Clifford monoid is semisimple (resp. von Neumann regular) if and only if the semigroup is a finite group and the characteristic p of k does not divide the order of this group.

Quantum doubles from a class of noncocommutative weak Hopf algebras

The concept of biperfect (noncocommutative) weak Hopf algebras is introduced and their properties are discussed. A new type of quasi-bicrossed products is constructed by means of weak Hopf skew-pairs of the weak Hopf algebras which are generalizations of the Hopf pairs introduced by Takeuchi. As a special case, the quantum double of a finite dimensional biperfect (noncocommutative) weak Hopf algebra is built. Examples of quantum doubles from a Clifford monoid as well as a noncommutative and noncocommutative weak Hopf algebra are given, generalizing quantum doubles from a group and a noncommutative and noncocommutative Hopf algebra, respectively. Moreover, some characterizations of quantum doubles of finite dimensional biperfect weak Hopf algebras are obtained.

The Right Braids, Quasi-braided Pre-tensor Categories, and General Yang–Baxter Operators#

This work is a development of braids, tensor categories and Yang–Baxter operators. According to Li [Li, F. (1998). Weak Hopf algebras and some new solution of quantum Yang-Baxter equation. J. Algebra 208:72–100; Li, F. (2000). Solutions of Yang-Baxter equation in endomorphism semigroup and quasi-(co)braided almost bialgebras. Comm. Algebra 28(5):2253–2270], it can be seen as a continuation of studying (not necessarily invertible) solutions of the (quantum) Yang–Baxter equation. We firstly introduce the right braid monoids and discuss their properties. Then, we define pre-tensor categories, pre-tensor functors and quasi-braided pre-tensor categories, and investiage their characterizations. Three examples are given from respectively a weak Hopf algebra, a crossed S-set of a Clifford monoid and the (strict) right braid category. Two universalities of the (strict) right braid category are gotten in order to characterize a category of general Yang–Baxter operators and a quasi-braided pre-tensor category. In a pre-tensor category we build a general centre of a pre-tensor category as a generalization of a centre and show that it is a quasi-braided pre-tensor category. At the end, a categorical interpretation of the quantum quasi-double of a weak Hapf algebra is obtained under a certain condition.

Inverse automata and monoids and the undecidability of the cayley subgraph problem for groups

The structure of an inverse monoid can be determined by the complete set of Schützenberger graphs of a presentation. Necessary and sufficient conditions for a collection of inverse X-graphs to be the complete set of Schützenberger graphs of some inverse monoid presentation are established and decidability results are obtained. Conditions for a single inverse X-graph to be a Schu¨tzenberger graph for some presentation are also obtained, and both problems are restricted to the case of Clifford monoids and E-unitary inverse monoids. Decidability and undecidability results are obtained for the case of finite graphs. It is also proved that the problem of embedding a finite inverse X-graph in the Cayley graph of a group is undecidable.