Idempotent Semigroups Research Articles

Some new properties on N semigroups

In this study we first show that  satisfies two important homological properties, namely Rees short exact sequence an d short five lemma. In addition, by defining inversive semigroup varieties of  we prove that strictly inverse semigroup  is isomorphic to the spined product of (C)-inversive semigroup and the idempotent semigroup of . Moreover, we give some consequences of the results to make a detailed classification over . It has been recently defined a new semigroup  based on Rees matrix and completely 0-simple semigroups. Further, it has been also proved finiteness conditions and the existence of some fundamental properties over .

Decomposition of idempotent pseudo-uninorms via ordinal sum

The decomposition of idempotent pseudo-uninorms is investigated. We show that each idempotent pseudo-uninorm on the unit interval can be decomposed into an ordinal sum of trivial semigroups and non-commutative idempotent semigroups defined on two elements, where the corresponding semigroup operation is the projection to one of the coordinates. Linear orders yielding idempotent pseudo-uninorms via ordinal sum of this type of semigroups are also investigated. The link between linear orders corresponding to an idempotent pseudo-uninorm and its dual pseudo-uninorm is shown for two types of duality.

The Source of Semiprimeness of Semigroups

In this study, we define new semigroup structures using the set S S = a ∈ S | a S a = 0 which is called the source of semiprimeness for a semigroup S with zero element. S S − idempotent semigroup, S S − regular semigroup, S S − reduced semigroup, and S S − nonzero divisor semigroup which are generalizations of idempotent, regular, reduced, and nonzero divisor semigroups in semigroup theory are investigated, and their basic properties are determined. In addition, we adapt some well-known results in semigroup theory to these new semigroups.

Characterization of n-uninorms with continuous underlying functions via z-ordinal sum construction

The n-uninorms with continuous underlying t-norms and t-conorms are characterized via the z-ordinal sum construction. We show that each n-uninorm with continuous underlying t-norms and t-conorms can be expressed as a z-ordinal sum of a countable number of Archimedean and idempotent semigroups with respect to the branching set A∼{z1,…,zn−1}, where the corresponding partial order has a tree structure.

Commutative semigroups with bounded orders of subdirectly irreducible acts

Subdirectly irreducible universal algebras, i.e., algebras which are not decomposable into non-trivial subdirect product, play an important role in mathematics due to well-known Birkhoff Theorem which states that any algebra is a subdirect product of subdirectly irreducible algebras (in another terminology: every algebra is approximated by the subdirectly irreducible algebras). In view of this, it seems reasonable to study algebras with certain conditions on subdirectly irreducible algebras. One of the natural restrictions is the finiteness of all subdirectly irreducible algebras. More stronger restriction is boundness of orders of all subdirectly irreducible algebras. An act over a semigroup (it is also an automaton, and a unary algebra) is a set with an action of the given semigroup on it. The acts over a fixed semigroup form a variety whose signature coincides with self semigroup. On the other hand, it is a category whose morphisms are homomorphisms from one act into another. It is not difficult to see that the semigroups over which all subdirectly ireducible acts are finite, are exactly the semigroups over which all subdirectly irreducible acts are finitely approximated (in another terminology: residually finite). A more narrow class form semigroups over which all acts are approximated by acts of n or less elements where n is a fixed natural number. In 2000, I.B.Kozhukhov proved that all non-trivial acts over a semigroup S are approximated by two-element ones if and only if S is a semilattice (a commutative idempotent semigroup). In 2014, I.B.Kozhukhov and A.R.Haliullina proved that any semigroup with bounded orders of dubdirectly irreducible acts is uniformly locally finite, i.e., for every k, the orders of k-generated subsemigroups are bounded. In the work of I.B.Kozhukhov and A.V.Tsarev 2019, the authors described completely the abelian groups over which all acts are finitely approximated, and also abelian groups over which all acts are approximated by acts of bounded orders. In this work, we characterize the commutative semigroups over which all acts are approximated by acts consisted of n or less elements.

Characterization of idempotent n-uninorms

The structure of idempotent n-uninorms is studied. We show that each idempotent 2-uninorm can be expressed as an ordinal sum of an idempotent uninorm (possibly also of a countable number of idempotent semigroups with operations min and max) and a 2-uninorm from Class 1 (possibly restricted to open or half-open unit square). Similar results are shown also for idempotent n-uninorms. Further, it is shown that idempotent n-uninorms are in one-to-one correspondence with special lower semi-lattices defined on the unit interval. The z-ordinal sum construction for partially ordered semigroups is also defined.

On the spectra of commutative semigroups

This study aims to investigate the Zariski topology on the prime ideals of a commutative semigroup S, denoted by Spec(S). First, we show that a topological space X is homeomorphic to Spec(S) for some commutative semigroup S if and only if X is an SS-space that can be described purely in topological terms. Next, we show that an adjunction exists between the category of commutative semigroups and that of SS-spaces. We further show that the category of commutative idempotent semigroups is dually equivalent to that of SS-spaces.

Planar semilattices and nearlattices with eighty-three subnearlattices

Finite (upper) nearlattices are essentially the same mathematical entities as finite semilattices, finite commutative idempotent semigroups, finite join-enriched meet semilattices, and chopped lattices. We prove that if an nelement nearlattice has at least 83 · 2 n−8 subnearlattices, then it has a planar Hasse diagram. For n > 8, this result is sharp.

Подсистемы и автоморфизмы некоторых конечных магм порядка k + k2

This work is devoted to the study of subsystems of some finite magmas S = (V, ∗) with a generating set of k elements and order k + k2. For k > 1, the magmas S are not semigroups and quasigroups. An element-by-element description of all magmas S subsystems is given. It was found that all the magmas S have subsystems that are semigroups. For k > 1, subsystems that are idempotent nonunit semigroups are explicitly indicated. Previously, a description of an automorphism group was obtained for magmas S. In particular, every symmetric permutation group Sk is isomorphic to the group of all automorphisms of a suitable magma S. In this paper, a general form of automorphism for a wider class of finite magmas of order k + k2 is obtained.

Finite bands are finitely related

We prove that every finite idempotent semigroup (band) is finitely related, which means that the clone of its term operations (i.e. operations induced by words) is determined by finitely many relations. This solves an open problem posed by Peter Mayr in 2013.

The subpower membership problem for bands

Fix a finite semigroup S and let a1,…,ak,b be tuples in a direct power Sn. The subpower membership problem (SMP) for S asks whether b can be generated by a1,…,ak. For bands (idempotent semigroups), we provide a dichotomy result: if a band S belongs to a certain quasivariety, then ▪ is in P; otherwise it is NP-complete.Furthermore we determine the greatest variety of bands all of whose finite members induce a tractable ▪. Finally we present the first example of two finite algebras that generate the same variety and have tractable and NP-complete SMPs, respectively.

Global determinism of idempotent semigroups

ABSTRACTThe power semigroup, or global, of a semigroup S is the set 𝒫(S) of all nonempty subsets of S equipped with the naturally defined multiplication. A class 𝒦 of semigroups is globally determined if any two members of 𝒦 with isomorphic globals are themselves isomorphic. The principal goal of this paper is to prove that the class of all idempotent semigroups is globally determined.

On regularity and the word problem for free idempotent generated semigroups

The category of all idempotent generated semigroups with a prescribed structure $\mathcal{E}$ of their idempotents $E$ (called the biordered set) has an initial object called the free idempotent generated semigroup over $\mathcal{E}$, defined by a presentation over alphabet $E$, and denoted by $\mathsf{IG}(\mathcal{E})$. Recently, much effort has been put into investigating the structure of semigroups of the form $\mathsf{IG}(\mathcal{E})$, especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for $\mathsf{IG}(\mathcal{E})$. We prove two principal results, one positive and one negative. We show that, for a finite biordered set $\mathcal{E}$, it is decidable whether a given word $w \in E^*$ represents a regular element; if in addition one assumes that all maximal subgroups of $\mathsf{IG}(\mathcal{E})$ have decidable word problems, then the word problem in $\mathsf{IG}(\mathcal{E})$ restricted to regular words is decidable. On the other hand, we exhibit a biorder $\mathcal{E}$ arising from a finite idempotent semigroup $S$, such that the word problem for $\mathsf{IG}(\mathcal{E})$ is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of $\mathsf{IG}(\mathcal{E})$ to the subgroup membership problem in finitely presented groups.

On endomorphism monoids in varieties of bands

It will be proved that every non-trivial variety \({\mathbb{V}}\) of bands (idempotent semigroups) contains a proper generating class of non-isomorphic bands B such that B generates \({\mathbb{V}}\) and any band \({B\prime}\) having the same endomorphism monoid as B is isomorphic to B or to the opposite band Bop. Consequently, every sharply greater band variety has a sharply greater class of endomorphism monoids.

Globals of Idempotent Semigroups

If S is a semigroup, the global (or the power semigroup) of S is the set P(S) of all nonempty subsets of S equipped with a naturally defined multiplication. A class K of semigroups is globally determined if any two semigroups of K with isomorphic globals are themselves isomorphic. We study properties of globals of idempotent semigroups and show, in particular, that the class of normal bands is globally determined.

Exact Unification and Admissibility

A new hierarchy of "exact" unification types is introduced, motivated by the study of admissible rules for equational classes and non-classical logics. In this setting, unifiers of identities in an equational class are preordered, not by instantiation, but rather by inclusion over the corresponding sets of unified identities. Minimal complete sets of unifiers under this new preordering always have a smaller or equal cardinality than those provided by the standard instantiation preordering, and in significant cases a dramatic reduction may be observed. In particular, the classes of distributive lattices, idempotent semigroups, and MV-algebras, which all have nullary unification type, have unitary or finitary exact type. These results are obtained via an algebraic interpretation of exact unification, inspired by Ghilardi's algebraic approach to equational unification.

Aggregation on Boolean multi-polar space: Knowledge-based vs. category-based ordering

Three types of aggregation operators defined on a Boolean multi-polar space {0,1,…,m} are compared. Three different monotonicity conditions that yield different types of aggregation operators are discussed and the structure of these aggregation operators on {0,1,…,m} with additional properties of associativity and commutativity is studied for any m∈N. The commutative associative Boolean multi-polar aggregation operators are shown to coincide with Boolean multi-polar uninorms. The structure of commutative associative SL aggregation operators is shown to be dependent on the structure of the class of Abelian idempotent semigroups based on m+1 elements with an annihilator. We show that each commutative associative SL aggregation operator with annihilator 0 corresponds to a lower semi-lattice with bottom element 0. For m>2 the class of commutative associative C aggregation operators is proved to be equal to the intersection of the class of commutative associative Boolean multi-polar aggregation operators and the class of commutative associative SL aggregation operators. Examples of the three types of commutative associative aggregation operators are shown and discussed for m=1,2,3.

Continuous semigroup structures on $$\mathbb {R}$$ R , cancellative semigroups and bands

Up to isomorphism there is only one continuous group operation on \(\mathbb {R}\), as Aczel (Bull Soc Math Fr 76: 59–64, 1949) showed in 1949. In this paper we show that there are exactly three distinct continuous (weakly) cancellative semigroup structures on \(\mathbb {R}\) modulo isomorphism. On the other hand, there are many continuous non-cancellative semigroup structures on \(\mathbb {R}\). We classify all ordered continuous bands (idempotent semigroups) on \(\mathbb {R}\). There are exactly eight (five) distinct ordered continuous band structures on \(\mathbb {R}\) modulo isomorphism (and anti-isomorphism).

A note on an alternating upper bound for random walks on semigroups

We consider random walks on idempotent semigroups, called Left Regular Bands, satisfying the relation xyx=xy for any two elements x and y of the semigroup. We give an alternating upper bound for the total variation distance of a random walk on a Left Regular Band semigroup, improving the previous bound by Brown and Diaconis.

EVERY GROUP IS A MAXIMAL SUBGROUP OF THE FREE IDEMPOTENT GENERATED SEMIGROUP OVER A BAND

Given an arbitrary group G we construct a semigroup of idempotents (band) BGwith the property that the free idempotent generated semigroup over BGhas a maximal subgroup isomorphic to G. If G is finitely presented then BGis finite. This answers several questions from recent papers in the area.