Regular Monoid Research Articles

𝔐-Endoregular lattices

In a previous work, (dual)- m -Rickart lattices were studied. Now, in this paper, we introduce m -endoregular lattices as those lattices L such that m is a regular monoid, where m is a submonoid with zero of End lin ( L ) . We show that these lattices can be characterized in terms of m -Rickart and dual- m -Rickart lattices. Also, we compare these new lattices with those lattices in which every compact element is a complement. We characterize the m -endoregular lattices such that every idempotent in m is central in m and we show that for these lattices the complements are a sublattice which is a Boolean algebra. We introduce two new concepts, m - K -extending and m - T -lifting lattices. For these lattices, we show that the monoid m has a regular quotient monoid provided they satisfy m - C 2 and m - D 2 respectively. At the end, some applications to quotients categories of R-Mod are given.

On Study of Monoids in Which Right Ideals are Generators

The main purpose of this paper is to study some classes of monoids in which every right ideal is a generator. We describe commutative monoids with this property and specially we show that for a commutative monoid S, every ideal is a generator if and only if S is a cancellative hereditary monoid which is equivalent to $$S=G$$ or $$S=G\times A$$ where G is a group and A is the additive monoid of non-negative integers. Also it is shown that for a non-trivial commutative monoid S without zero any ideal is a generator if and only if there exists a free S-act A such that for any subact B of $$A, trace~(B,A)=A$$ . Moreover for a regular monoid S it is shown any right ideal of S is a generator if and only if S has no two sided ideal.

The category of affine algebraic regular monoids

<abstract><p>The main aim of this study is to characterize affine weak $ k $-algebra $ H $ whose affine $ k $-variety $ S = M_{k}(H, k) $ admits a regular monoid structure. As preparation, we determine some results of weak Hopf algebras morphisms, and prove that the anti-function from the category $ \mathcal{C} $ of weak Hopf algebras whose weak antipodes are anti-algebra morphisms is adjoint. Then, we prove the main result of this study: the anti-equivalence between the category of affine algebraic $ k $-regular monoids and the category of finitely generated commutative reduced weak $ k $-Hopf algebras.</p></abstract>

Endomorphisms of Clifford semigroups with injective structure homomorphisms

In the present paper, we study semigroups of endomorphisms on Clifford semigroups with injective structure homomorphisms, where the semilattice has a least element. We describe such Clifford semigroups having a regular endomorphism monoid. If the endomorphism monoid on the Clifford semigroup is completely regular then the corresponding semilattice has at most two elements. We characterize all Clifford semigroups Gα∪Gβ (α>β) with an injective structure homomorphism, where Gα has no proper subgroup, such that the endomorphism monoid is completely regular. In particular, we consider the case that the structure homomorphism is bijective.

Construction of Inverse Unit Regular Monoids from a Semilattice and a Group

This paper is a continuation of a previous paper [6] in which the structure of certain unit regular semigroups called R-strongly unit regular monoids has been studied. A monoid S is said to be unit regular if for each element s Î S there exists an element u in the group of units G of S such that s = sus. Hence where su is an idempotent and is a unit. A unit regular monoid S is said to be a unit regular inverse monoid if the set of idempotents of S form a semilattice. 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. Here we give a detailed study of inverse unit regular monoids and the results are mainly based on [10]. The relations between the semilattice of idempotents and the group of units in unit regular inverse monoids are better identified in this case. .

Sandwich semigroups in locally small categories I: foundations

Fix (not necessarily distinct) objects i and j of a locally small category S, and write $$S_{ij}$$ for the set of all morphisms $$i\rightarrow j$$ . Fix a morphism $$a\in S_{ji}$$ , and define an operation $$\star _a$$ on $$S_{ij}$$ by $$x\star _ay=xay$$ for all $$x,y\in S_{ij}$$ . Then $$(S_{ij},\star _a)$$ is a semigroup, known as a sandwich semigroup, and denoted by $$S_{ij}^a$$ . This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green’s relations and stability, focusing on the relationships between these properties on $$S_{ij}^a$$ and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set $${\text {Reg}}(S_{ij}^a)$$ of all regular elements of $$S_{ij}^a$$ is a subsemigroup of $$S_{ij}^a$$ . Under this condition, we carefully analyse the structure of the semigroup $${\text {Reg}}(S_{ij}^a)$$ , relating it via pullback products to certain regular subsemigroups of $$S_{ii}$$ and $$S_{jj}$$ , and to a certain regular sandwich monoid defined on a subset of $$S_{ji}$$ ; among other things, this allows us to also describe the idempotent-generated subsemigroup $$\mathbb E(S_{ij}^a)$$ of $$S_{ij}^a$$ . We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups $$S_{ij}^a$$ , $${\text {Reg}}(S_{ij}^a)$$ and $$\mathbb E(S_{ij}^a)$$ ; we give lower bounds for these ranks, and in the case of $${\text {Reg}}(S_{ij}^a)$$ and $$\mathbb E(S_{ij}^a)$$ show that the bounds are sharp under a certain condition we call MI-domination. Applications to concrete categories of transformations and partial transformations are given in Part II.

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.

Axiomatizability of the class of weakly injective S-acts

The concepts of weakly injective, fg-weakly injective, and p-weakly injective S-acts generalize that of injective S-act. We study the monoids S over which the classes of weakly injective, fg-weakly injective, and p-weakly injective S-acts are axiomatizable. We prove that the class of p-weakly injective S-acts over a regular monoid is axiomatizable.

Maximal subsemigroups and finiteness conditions on transformation semigroups with fixed sets

Let $Y$ be a fixed subset of a nonempty set $X$ and let $Fix(X,Y)$ be the set of all self maps on $X$ which fix all elements in $Y$. Then under the composition of maps, $Fix(X,Y)$ is a regular monoid. In this paper, we prove that there are only three types of maximal subsemigroups of $Fix\left(X,Y\right)$ and these maximal subsemigroups coincide with the maximal regular subsemigroups when $X\setminus Y$ is a finite set with $|X\setminus Y|\geq 2$. We also give necessary and sufficient conditions for $Fix(X,Y)$ to be factorizable, unit-regular, and directly finite.

End-regularity of zero-divisor graphs of group rings

ABSTRACTA graph Γ is said to be End-regular if its endomorphism monoid End(Γ) is regular. D. Lu and T. Wu [25] posed an open problem: Given a ring R, when does the zero-divisor graph Γ(R) have a regular endomorphism monoid? and they solved the problem for R a commutative ring with at least one nontrivial idempotent. In this paper, we solve this problem for zero-divisor graphs of group rings.

Multiplier Hopf Monoids

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele's definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved --- in an appropriate, multiplier-valued sense --- which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved --- which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.

Unicyclic graphs with regular endomorphism monoids

The motivation of this paper comes from an open question: which graphs have regular endomorphism monoids? In this paper, we give a definitely answer for unicyclic graphs, proving that a unicyclic graph [Formula: see text] is End-regular if and only if, either [Formula: see text] is an even cycle with 4, 6 or 8 vertices, or [Formula: see text] contains an odd cycle [Formula: see text] such that the distance of any vertex to [Formula: see text] is at most 1, i.e., [Formula: see text]. The join of two unicyclic graphs with a regular endomorphism monoid is explicitly described.

Natural Partial Orders on Transformation Semigroups with Fixed Sets

LetXbe a nonempty set. For a fixed subsetYofX, letFixX,Ybe the set of all self-maps onXwhich fix all elements inY. ThenFixX,Yis a regular monoid under the composition of maps. In this paper, we characterize the natural partial order onFix(X,Y)and this result extends the result due to Kowol and Mitsch. Further, we find elements which are compatible and describe minimal and maximal elements.

Homotopy bases and finite derivation type for subgroups of monoids

Given a monoid defined by a presentation, and a homotopy base for the derivation graph associated to the presentation, and given an arbitrary subgroup of the monoid, we give a homotopy base (and presentation) for the subgroup. If the monoid has finite derivation type (FDT), and if under the action of the monoid on its subsets by right multiplication the strong orbit of the subgroup is finite, then we obtain a finite homotopy base for the subgroup, and hence the subgroup has FDT. As an application we prove that a regular monoid with finitely many left and right ideals has FDT if and only if all of its maximal subgroups have FDT. We use this to show that a finitely presented regular monoid with finitely many left and right ideals satisfies the homological finiteness condition FP3 if all of its maximal subgroups satisfy the condition FP3.

On residual finiteness of monoids, their Schützenberger groups and associated actions

In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M; (RFSG) residual finiteness of Schützenberger groups of M; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table.

Centralizer of an idempotent in a reductive monoid

Abstract Let M be a reductive monoid. If e is an idempotent in M, we prove that the centralizer M(e) of e in M is a regular monoid with a finite graded poset of 𝒥-classes. We compute this poset explicitly when M is of canonical or dual canonical type and e is the relevant pivotal idempotent.

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].

Kernels, regularity and unipotent radicals in linear algebraic monoids

An (affine) algebraic monoid is an affine variety over an algebraically closed field K endowed with a monoid structure such that the product map is an algebraic variety morphism. Let M be an irreducible algebraic monoid with G (⊊ M) its unit group, ker(M) its semigroup kernel, E(ker(M)) the set of minimal idempotents of M. Algebraically, ker(M) is the minimum regular 𝒥-class in M; geometrically, ker(M) is a retract of M but also the unique closed orbit under the natural G × G-action on M. We study regularity conditions for M and the relationships between ker(M) and Ru(G). We also study an algebraic group embedding problem. The principal results are: (i) dim Ru(G) = dim E(ker(M)) + dim Ru(H) + dim Ru(G(e)), where e ∈ E(ker(M)), H is a maximal subgroup of ker(M) and G(e) ≔ {x ∈ G | xe = ex = e}. (ii) A connected linear algebraic group with nontrivial characters can be realized as a proper unit group of some irreducible normal regular algebraic monoid. (iii) When char (K) = 0, if M is regular, dim Ru(G) = dim ker(M) implies Ru(G) ≅ ker(M) as algebraic varieties.

CODING PARTITIONS OF REGULAR SETS

A coding partition of a set of words partitions this set into classes such that whenever a sequence, of minimal length, has two distinct factorizations, the words of these factorizations belong to the same class. The canonical coding partition is the finest coding partition that partitions the set of words in at most one unambiguous class and other classes that localize the ambiguities in the factorizations of finite sequences. We prove that the canonical coding partition of a regular set contains a finite number of regular classes and we give an algorithm for computing this partition. From this we derive a canonical decomposition of a regular monoid into a free product of finitely many regular monoids.

On endomorphism-regularity of zero-divisor graphs

The paper studies the following question: Given a ring R , when does the zero-divisor graph Γ ( R ) have a regular endomorphism monoid? We prove if R contains at least one nontrivial idempotent, then Γ ( R ) has a regular endomorphism monoid if and only if R is isomorphic to one of the following rings: Z 2 × Z 2 × Z 2 ; Z 2 × Z 4 ; Z 2 × ( Z 2 [ x ] / ( x 2 ) ) ; F 1 × F 2 , where F 1 , F 2 are fields. In addition, we determine all positive integers n for which Γ ( Z n ) has the property.