Semigroups

In the history of mathematics, the algebraic theory of semigroups is a relative new-comer, with the theory proper developing only in the second half of the twentieth century. Before this, however, much groundwork was laid by researchers arriving at the study of semigroups from the directions of both group and ring theory. In this paper, we will trace some major strands in the early development of the algebraic theory of semigroups. We will begin with the aspects of the theory which were directly inspired by, and were analogous to, existing results for both groups and rings, before moving on to consider the first independent theorems on semigroups: theorems with no group or ring analogues.

ABSTRACT A filter in a semigroup is a subsemigroup whose complement is an ideal. (Alternatively, in a quasiordered semigroup, a slightly more general definition can be given.) We prove a number of results related to filters in a semigroup and the lattice of filters of a semigroup. For instance, we prove that every complete algebraic lattice can be the lattice of filters of a semigroup. We prove that every finite semigroup is a homomorphic image of a finite semigroup whose lattice of filters is boolean and which belongs to the pseudovariety generated by the original semigroup. We describe filter lattices of some well-known semigroups such as full transformation semigroups of finite sets (which are three-element chains) and free semigroups (which are boolean).

Abstract It is well known that the smallest semilattice congruence can be described via filters. We generalise this result to the smallest left (right) normal band congruences and also to arbitrary semilattice (left normal band, right normal band) congruences, describing them ...

We study left and right negatively orderable semigroups, natural one-sided generalizations of negatively orderable semigroups, in the wide class $${\mathfrak {C}}$$C of semigroups where every element has a left and a right identity. In semigroups in $${\mathfrak {C}}$$C the smallest left and smallest right negatively orderable congruences $$\theta _{{ }_{{\mathrm {LNO}}}}$$θLNO and $$\theta _{{ }_{{\mathrm {RNO}}}}$$θRNO commute; furthermore, we have: $$\theta _{{ }_{{\mathrm {LNO}}}}\circ \theta _{{ }_{{\mathrm {RNO}}}}=\theta _{{ }_{{\mathrm {RNO}}}}\circ \theta _{{ }_{{\mathrm {LNO}}}}=\theta _{{ }_{{\mathrm {RNO}}}}\vee \theta _{{ }_{{\mathrm {LNO}}}}=\theta _{{ }_{{\mathbf {{\mathrm {NO}}}}}}$$θLNO∘θRNO=θRNO∘θLNO=θRNO∨θLNO=θNO, where $$\theta _{{ }_{{\mathbf {{\mathrm {NO}}}}}}$$θNO is the smallest negatively orderable congruence. These properties give rise to the following decomposition: for every semigroup S in $${\mathfrak {C}}$$C, $$S/(\theta _{{ }_{{\mathrm {LNO}}}}\cap \theta _{{ }_{{\mathrm {RNO}}}})$$S/(θLNO∩θRNO) is isomorphic to the spined product of the largest left and the largest right negatively orderable images $$S/\theta _{{ }_{{\mathrm {LNO}}}}$$S/θLNO and $$S/\theta _{{ }_{{\mathrm {RNO}}}}$$S/θRNO of S over its largest negatively orderable image $$S/\theta _{{ }_{{\mathbf {{\mathrm {NO}}}}}}$$S/θNO. Descriptions of finite left (right) negatively orderable semigroups and of $$(\theta _{{}_{{\mathrm {LNO}}}}\cap \theta _{{}_{{\mathrm {RNO}}}})$$(θLNO∩θRNO)-trivial semigroups in $${\mathfrak {C}}$$C are obtained and the pseudovarieties generated by these semigroups are determined. A regular semigroup is left (right) negatively orderable if and only if it is a right (left) normal band. Counterexamples of semigroups where $$\theta _{{ }_{{\mathrm {LNO}}}}\circ \theta _{{ }_{{\mathrm {RNO}}}} \ne \theta _{{ }_{{\mathrm {RNO}}}}\circ \theta _{{ }_{{\mathrm {LNO}}}}$$θLNO∘θRNO≠θRNO∘θLNO and $$\theta _{{ }_{{\mathrm {LNO}}}}\vee \theta _{{ }_{{\mathrm {RNO}}}} \ne \theta _{{ }_{{\mathbf {{\mathrm {NO}}}}}}$$θLNO∨θRNO≠θNO are presented.

The reader is guided through a detailed proof of the Ehresmann-Schein-Nambooripad Theorem from the point of view of symmetry, following Lawson's proof in Inverse Semigroups: The Theory of Partial Symmetries. Sierpiński's Sieve is taken as an example. Arrow diagrams, natural orders, restricted products, and prehomomor-phisms are discussed for inverse semigroups; properties of inductive groupoids and their pseudoproducts are established. There is a short interlude on programming, Sierpiński's Sieve, and quasicrystals. A couple of exercises are suggested.

We study the partial actions of monoids on sets, in the sense of Megrelishvili and Schröder, and derive generalisations of results already obtained in the group case by Kellendonk and Lawson. Particular attention is paid to the question of whether it is possible to construct (global) actions from such partial actions. We describe two methods for accomplishing this: the method by 'expansion' and that by 'globalisation'. The former is adapted directly from the group case, whilst the latter extends a result of Megrelishvili and Schröder.

We find that central to the study of partial monoid actions is a class ofsemigroups termed 'weakly left E-ample semigroups'. These generalise inverse semigroups and arise very naturally as subsemigroups of partial transformation monoids which are closed under an additional unary operation +. With the desired results established for arbitrary monoids, we next adapt these to the semigroup case in the specific instance of weakly left E-ample semigroups, that is, we adapt our earlier results to take account of the presence of the + operation. These latter results yield specialisations to the inverse case.

We finally introduce the notion of an 'inductive E-constellation' and show that such objects correspond to weakly left E-ample semigroups in a manner analogous to the correspondence between inductive groupoids and inverse semigroups. We study the actions and partial actions of inductive E-constellations on sets and connect these with our previous results for weakly left E-ample semigroups.

Introduced several new axiomatic systems, that are not less general than group theory, and discovered discontinuous analysis.

In this work I introduce and study in details the concepts of funcoids
which generalize proximity spaces and reloids which generalize uniform
spaces, and generalizations thereof. The concept of funcoid is generalized
concept of proximity, the concept of reloid is cleared from superfluous
details (generalized) concept of uniformity.

Also funcoids and reloids are generalizations of binary relations
whose domains and ranges are filters (instead of sets). Also funcoids
and reloids can be considered as a generalization of (oriented) graphs,
this provides us with a common generalization of calculus and discrete
mathematics.

I consider (generalized) limit of arbitrary (discontinuous) function, defined in terms of funcoids.
Definition of generalized limit makes it obvious to define such things as derivative of an arbitrary function, integral of an arbitrary function, etc. It is given a definition of non-dif\-fe\-re\-nti\-able solution of a (partial) differential equation. It's raised the question how do such solutions ``look like'' starting a possible big future research program.

The generalized solution of one simple example differential equation is also considered.

The generalized derivatives and integrals are linear operators. For example $\int_a^b f(x)dx - \int_a^b f(x)dx = 0$ is defined and true for \emph{every} function.

The concept of continuity is defined by an algebraic formula (instead
of old messy epsilon-delta notation) for arbitrary morphisms (including
funcoids and reloids) of a partially ordered category. In one formula
continuity, proximity continuity, and uniform continuity are generalized.

Also I define connectedness for funcoids and reloids.

Then I consider generalizations of funcoids: pointfree funcoids and
generalization of pointfree funcoids: staroids and multifuncoids.
Also I define several kinds of products of funcoids and other morphisms.

I define \emph{space} as an element of an ordered semigroup action, that is a semigroup action conforming to a partial order. Topological spaces, uniform spaces, proximity spaces, (directed) graphs, metric spaces, etc.\ all are spaces. It can be further generalized to ordered precategory actions (that I call interspaces). I build basic general topology (continuity, limit, openness, closedness, hausdorffness, compactness, etc.)\ in an arbitrary space. Now general topology is an algebraic theory.

Before going to topology, this book studies properties of co-brouwerian
lattices and filters.

The purpose of this paper is to investigate, the characterizations of different classes of non-associative ordered semigroups by using anti fuzzy left (resp. right, interior) ideals. 2020 Mathematics Subject Classifications: 13Cxx, 94D05, 13Axx, 18B40

Let X be a nonempty set. The full transformation, T (X) on a set X is the mapping from X into itself with a composition operation. This study is sequel to the work of Mendes-Goncalves and Sullivan (2011), [9] who studied the semigroup of transformations restricted by an equivalence E(X, σ). We discuss some structures of the semigroup of transformations restricted by an equivalence , E(X, σ) considering it on two classes of semigroups namely, Complete regular semigroup and Inverse Semigroup and showed that E(X, σ) is completely regular but not an inverse semigroup on its regular part-the largest regular subsemigroup E for any nontrivial set X and characterize its Starred Ideals. Mathematics Subject Classification (2010). 20M20

Semigroup theory is a thriving field in modern abstract algebra, though perhaps not a very well-known one. In this article, we give a brief introduction to the theory of algebraic semigroups and hopefully demonstrate that it has a flavor quite different from that of group theory. In the first few sections, we build up some basic semigroup theory, always with the group analogy at the back of our minds. Then, once we have established enough theory, we break free of this restriction and see some truly “independent” semigroup theory. In the final section, we consider the application of semigroups to the study of “partial symmetries.”

We consider the investigation of the embedding of semigroups in groups, a problem which spans the early-twentieth-century development of abstract algebra.  Although this is a simple problem to state, it has proved rather harder to solve, and its apparent simplicity caused some of its would-be solvers to go awry.  We begin with the analogous problem for rings, as dealt with by Ernst Steinitz, B. L. van der Waerden and Øystein Ore.  After disposing of A. K. Sushkevich's erroneous contribution in this area, we present A. I. Maltsev's example of a cancellative semigroup which may not be embedded in a group, which showed for the first time that such an embedding is not possible in general.  We then look at the various conditions that were derived for such an embedding to take place: the sufficient conditions of Paul Dubreil and others, and the necessary and sufficient conditions obtained by A. I. Maltsev, Vlastimil Pták and Joachim Lambek.  We conclude with some comments on the place of this problem within the theory of semigroups, and also within abstract algebra more generally.

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an \emph{Armendariz map} between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.

One of the major issues facing mathematics at present is the demand for immediate applications. Another (historical) example of such a policy may be found in Soviet ideological interference in mathematics, of which I will give a short description, before illustrating it with an account of an ideological attack on the Russian algebraist E. S. Lyapin.

The aim of this literature study is to provide a theoretical basis for “preliminary” concepts as set forth in the article "On generalized inverses and Green’s relations" by X.Mary, with a focus on its application to matrices over an arbitrary field specifically. It could then serve as a reference, or aid, in further research related to the specific article and subject, even if only to identify questions, or possible further topics within the field that might be worth researching. So the goal is to expand and explain the concepts, and also how it specifically relates to matrix theory and generalized matrix inverses. The structure of this study will naturally follow the structure of the main reference article closely, and so it is recommended to have a copy of the reference article in hand while studying this article. When referring to matrices it will always be matrices with finite dimension over a (arbitrary) field.

This is the second in a projected series of three papers, the aim of which is the complete description of the closure of any one-parameter inverse semigroupin a locally compact topological inverse semigroup. In it we characterize all one-parameter inverse semigroups. In order to accomplish this, we construct the free
one-parameter inverse semigroups and then describe their congruences.

Anton Kazimirovich Suschkewitsch was a Russian mathematician who spent most of his working life at Kharkov State University in the Ukraine. In the 1920s, he embarked upon the first systematic study of semigroups, placing him at the very beginning of algebraic semigroup theory and, arguably, earning him the title of the world’s first semigroup theorist. Owing to the political circumstances under which he lived, however, his work failed to find a wide audience during his lifetime. We give a brief account of his life and his researches into semigroup theory.

In this paper we give a full description of idempotent elements of the semigroup BX (D), which are defined by semilattices of the class Σ1 (X, 10). For the case where X is a finite set we derive formulas by means of which we can calculate the numbers of idempotent elements of the respective se-migroup.

In this paper, let be a nite set, be a complete-semilattice of unions and = {{ 1 \ 1 = , 1 \ 2 = , 4 \ 3 = , 3 \ 4 = , 6 \ 7 = , 7 \ 6 = , 2 ∪ 1 = 3 , 4 ∪ 3 = 5 , 6 ∪ 7 = 8. Using the characteristic family of sets, the characteristic mapping and base sources of , we characterize the class whose elements are each isomorphic to. We generate some advanced formulas in order to calculate the number of regular elements of (() satisfying ((,) = , in an eecient way.

In this paper let Q = {T1, T2, T3, T4, T5, T6, T7, T8} be a subsemilattice
of X−semilattice of unions D where T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T6 ⊂ T8,
T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T7 ⊂ T8, T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T6 ⊂ T8, T1 ⊂ T2 ⊂
T4 ⊂ T5 ⊂ T7 ⊂ T8, T1 6= ∅, T4\T3 6= ∅, T3\T4 6= ∅, T6\T7 6= ∅, T7\T6 6= ∅,
T3 ∪T4 = T5, T6 ∪T7 = T8, then we characterize the class each element of which
is isomorphic to Q by means of the characteristic family of sets, the characteristic
mapping and the generate set of Q. Moreover, we calculate the number of
regular elements of BX(D) for a finite set X.

The Ehresmann-Schein-Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended the ESN Theorem to the classes of ample, weakly ample and weakly $E$-ample semigroups. A semigroup in any of these classes must contain a semilattice of idempotents, but need not be regular. It is significant here that these classes are each defined by a set of conditions and their left-right duals.

Recently, a class of semigroups has come to the fore that is a one-sided version of the class of weakly E-ample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of partial semigroups we dub inductive constellations, together with the appropriate notion of ordered map, which we call inductive radiant. We note that such objects have appeared outside of semigroup theory in the work of Exel. In a subsequent article we develop a theory of partial action and expansion for inductive constellations, along the lines of that of Gilbert for inductive groupoids.

In this paper we study flatness properties (pullback flatness, limit flatness, finite limit flatness) of acts over semigroups. These are defined by requiring preservation of certain limits from the functor of tensor multiplication by a given act. We give a description of firm pullback flat acts using Conditions (P) and (E). We also study pure epimorphisms and their connections to finitely presented acts and pullback flat acts. We study these flatness properties in the category of all acts, as well as in the category of unitary acts and in the category of firm acts, which arise naturally in the Morita theory of semigroups.

Difficulties encountered in studying generators of semigroup () X B D of binary relations defined by a complete X-semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup () X B D , which are defined by semilattice D, can be generated by the set () () { } X B B D V X D α α * = ∈ = , .

In this paper, we take Q16 subsemilattice of D and we will calculate the number of right unit, idem-potent and regular elements α of BX (Q16) satisfied that V (D, α) = Q16 for a finite set X. Also we will give a formula for calculate idempotent and regular elements of BX (Q) defined by an X-semilattice of unions D.

For any group G and set A, a cellular automaton over G and A is a transformation from A^G to A^G defined via a finite neighborhood (called a memory set) and a local function. In this paper, we assume that G and A are both finite and study various algebraic properties of the finite monoid CA(G;A) consisting of all cellular automata over G and A. Let ICA(G;A) be the group of invertible cellular automata over G and A. In the first part, using information on the conjugacy classes of subgroups of G, we give a detailed description of the structure of ICA(G;A) in terms of direct and wreath products. In the second part, we study generating sets of CA(G;A). In particular, we prove that CA(G,A) cannot be generated by cellular automata with small memory set, and, when G is finite abelian, we determine the minimal size of a subset V of CA(G;A) such that CA(G;A) = < ICA(G;A), V >.

461 pages, 11 chapters.

The book is devoted to the thorough study of polyadic (higher arity) algebraic structures, which has a long history, starting from 19th century. The main idea was to take a single set, closed under one binary operation, and to 'generalize' it by increasing the arity of the operation, called a polyadic operation. Until now, a general approach to polyadic concrete many-set algebraic structures was absent. We propose to investigate algebraic structures in the 'concrete way' and provide consequent 'polyadization' of each operation, starting from group-like structures and finishing with the Hopf algebra structures. Polyadic analogs of homomorphisms which change arity, heteromorphisms, are introduced and applied for constructing unusual representations, multiactions, matrix representations and polyadic analogs of direct product. We provide the polyadic generalization of the Yang–Baxter equation, find its constant solutions, and introduce polyadic tensor categories. Suitable for university students of advanced level algebra courses and mathematical physics courses.

Key features
• Provides a general, unified approach • Widens readers perspective of the possibilities to develop standard algebraic structures
• Provides the new kind of homomorphisms changing the arity, heteromorphisms, are introduced and applied for construction of new representations, multiactions and matrix representations
• Presents applications of 'polyadization' approach to concrete algebraic structures.

For the light paperback version of the book use MyPrint service here: https://iopscience.iop.org/book/978-0-7503-2648-3 also PDF, ePub and Kindle.

For libraries and direct ordering from IOP: https://store.ioppublishing.org/page/detail/Polyadic-Algebraic-Structures//?k=9780750326483

Amazon.com ordering: https://www.amazon.com/Polyadic-Algebraic-Structures-Steven-Duplij/dp/0750326468

LinkedIn (with free Front Matter and 1st Chaper):  https://www.linkedin.com/posts/duplij_arity-algebraicstructure-field-activity-6940082232648720384-e2qk

Schutzenberger showed in 1965 that star-free rational languages are exactly those whose syntactic monoid is aperiodic (has only trivial subgroups). We describe the complexity hierarchy on finite aperiodic semigroups and on star-free languages, arising from the Krohn-Rhodes theorem. Using techniques of J. Almeida on the semi direct product of varieties and some results on idempotent semigroups, we show that this complexity hierarchy is effectively decidable and results in a strictly increasing filtration of the class of aperiodic semigroups. Using the notion of syntactic monoid and Kleene's Theorem, a corresponding result follows for star-free languages.

The Ehremann--Schein--Nambooripad Theorem expresses the fundamental connection between the notions of inverse semigroups and inductive groupoids, which exists because these concepts provide two distinct approaches to the study of one-one partial transformations.  In the case of arbitrary partial transformations, the analogous two approaches are provided by restriction semigroups and inductive categories, the former being generalisations of inverse semigroups, and the latter of inductive groupoids.  There is indeed also a generalisation of the Ehremann--Schein--Nambooripad Theorem which encapsulates the connection between these two more general objects.  In this article, we will explore the origins of these theorems, and survey the basic theory surrounding them.

We derive necessary and sufficient conditions for the Birget--Rhodes prefix expansion of a monoid to be (weakly) left ample, thereby proving analogues of the results already obtained for the related Szendrei expansion by Fountain, Gomes and Gould, and Fountain and Gomes. As a corollary, we obtain conditions for the prefix expansion to be inverse.

A BCK-algebra is an algebraic structure of a set X with one binary operation. A KS-semigroup is a semigroup with respect to one binary operation and also a BCK-algebra with respect to another binary oper-ation satisfying left and right distributive laws. This paper characterizes KS-semigroup homomorphisms and proves the isomorphism theorems for KS-semigroups.

recently proved that there exists a metric semigroup U such that every compact metric semigroup with property P is topologically isomorphic to a subsemigroup of U ; where a semigroup S has property P if and only if for each x, y in S , x t y , there is a z in S such that xz # yz or zx t zy. A stronger result is proved here more simply. It is shown that for any set A of metric semigroups there exists a metric semigroup U such that each 5 in A is topologically isomorphic to a subsemigroup of U. In particular this is the case when A is the class of all separable metric semigroups, or more particularly the class of all compact metric semigroups. THEOREM. Let A be any set of metric semigroups. Then there exists a metric semigroup U suck that for each S in A there is a mapping of S into V which is both a semigroup isomorphism and a homeomorphism.

We study properties of particular right topological semigroups, namely the Stone-ˇ Cech compactifications of semilattices. In particular, we extend some of the results of Grainger's paper [3] on the Stone-ˇ Cech compact-ification of the semilattice of finite subsets of an infinite set. In his paper [3], A. D. Grainger considers the semigroup S = P f (J), where J is an infinite set, P f (J) denotes the set of all finite non-empty subsets of J, and P f (J) becomes a semigroup under the operation ∪ of set theoretic union. As for any semigroup S, the set βS of all ultrafilters on S is, in a natural way, a compact right topological semigroup; Grainger's paper studies some of the algebraic, topological, and topological-dynamical properties of βS. In the present paper, we give more straightforward proofs for some of his results, improve some, and show that some can be generalized to arbitrary semilattices. The standard text on the algebra and topology of the Stone-ˇ Cech compactifi-cation of an arbitrary discrete semigroup is [4]. However, we shall give, as far as possible, proofs which do not make use of the theory developed in that book. I thank A. D. Grainger for giving me a preprint of his paper and the referee for pointing out several flaws in the original version. 1. Semilattices We define in this section the notion of a semilattice and carry out for semilattices some of the theory begun in [3]. Let us point out that semigroups are always assumed to be non-empty. Definition 1.1. We call a semigroup (S, +) a semilattice (more exactly, an upper semilattice) if its addition + is commutative and idempotent, i.e. x + y = y + x and x + x = x hold for arbitrary x, y in S. Remark 1.2. On a semilattice (S, +), we have a natural partial ordering defined by x ≤ y ⇐⇒ x + y = y. With respect to this partial order, x + y is the least upper bound of x and y in S. Consequently, we will write x ∨ y for x + y. Conversely, if (S, ≤) is a partial order in which any two elements x, y have a least upper bound x ∨ y, then (S, ∨) is a semilattice. The class of semilattices is defined by identities (associativity, commutativity, idempotence), hence a variety in the sense of universal algebra. Thus arbitrary semilattices are the homomorphic images of free ones. These are easily described: they are simply the semilattices of the form (P f (J), ∪).

We investigate \emph{partial monoid actions}, in the sense of Megrelishvili and Schr\"{o}der. These are equivalent to a class of premorphisms, which we call \emph{strong} premorphisms. We describe two distinct methods for constructing a monoid action from a partial monoid action: the \emph{expansion} method provides a generalisation of a result of Kellendonk and Lawson in the group case, whilst the approach via \emph{globalisation} extends results of both Megrelishvili and Schr\"{o}der, and Kellendonk and Lawson.

The Ehresmann-Schein-Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended the ESN Theorem to the classes of ample, weakly ample and weakly $E$-ample semigroups. A semigroup in any of these classes must contain a semilattice of idempotents, but need not be regular. It is significant here that these classes are each defined by a set of conditions and their left-right duals. Recently, a class of semigroups has come to the fore that is a one-sided version of the class of weakly E-ample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of part...

We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents.

In this paper let Q = {T1, T2, T3, T4, T5, T6, T7, T8} be a subsemilattice of X−semilattice of unions D where T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T6 ⊂ T8, T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T7 ⊂ T8, T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T6 ⊂ T8, T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T7 ⊂ T8, T1 6= ∅, T4\T3 6= ∅, T3\T4 6= ∅, T6\T7 6= ∅, T7\T6 6= ∅, T3 ∪T4 = T5, T6 ∪T7 = T8, then we characterize the class each element of which is isomorphic to Q by means of the characteristic family of sets, the characteristic mapping and the generate set of Q. Moreover, we calculate the number of regular elements of BX(D) for a finite set X.

By a topological semiring we mean a Hausdorff space S together
with two continuous associative operations on S such that one (called
multiplication) distributes across the other (called addition). That
is, we insist that x(y+z)=xy+xz and (x+y)z = xz+yz for all x, y,
and z in S. Note that, in contrast to the purely algebraic situation,
we do not postulate the existence of an additive identity which is a
multiplicative zero.
In this note we point out a rather weak multiplicative condition
under which each additive subgroup of a compact semiring is totally
disconnected. We also give several corollaries and examples.