N-ary Semigroups Research Articles

Semilattices of simple and regular n-ary semigroups

Semilattices of simple and regular n-ary semigroups

Prime i-Ideals in Ordered n-ary Semigroups

We study the concept of i-ideal of an ordered n-ary semigroup and give a construction of the i-ideal of an ordered n-ary semigroup generated by its nonempty subset. Moreover, we study the notions of prime, weakly prime, semiprime and weakly semiprime ideals of an ordered n-ary semigroup.

Reducibility of n-ary semigroups: from quasitriviality towards idempotency

Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying the condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ of the elements $x_1,\ldots,x_n$ are equal to each other. The elements of $\mathcal{F}^n_1$ are said to be quasitrivial and those of $\mathcal{F}^n_n$ are said to be idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathcal{F}^n_{n-1}\subseteq\mathcal{F}^n_n$ and we give conditions on the set $X$ for the last inclusions to be strict. The class $\mathcal{F}^n_1$ was recently characterized by Couceiro and Devillet, who showed that its elements are reducible to binary associative operations. However, some elements of $\mathcal{F}^n_n$ are not reducible. In this paper, we characterize the class $\mathcal{F}^n_{n-1}\setminus\mathcal{F}^n_1$ and show that its elements are reducible. We give a full description of the corresponding reductions and show how each of them is built from a quasitrivial semigroup and an Abelian group whose exponent divides $n-1$.

Post theorem for strongly dependent n-ary semigroups

Abstract We use an analogue of Gluskin–Hosszú theorem for strongly dependent finite n-ary semigroups to prove an analogue of Post coset theorem. We show that in fact these theorems are equivalent. Thus it turns our that the case of n-ary semigroups is fully similar to the case of n-groups.

Every quasitrivial n-ary semigroup is reducible to a semigroup

We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in the case of symmetric n-ary semigroups. We also explicitly determine the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences.

Analogues of Gluskin–Hosszú and Malyshev theorems for strongly dependent n-ary operations

Abstract The paper contains an extension of Malyshev theorem for n-ary quasigroups with a right or left weak invertibility property to the case of strongly dependent n-ary operations. As a corollary a new proof of Gluskin–Hosszú theorem for strongly dependent n-ary semigroups is obtained.

Свойства сильно зависимых n-арных полугрупп

Свойства сильно зависимых n-арных полугрупп

The Minimality and Maximality of n-ideals in n-ary Semigroups

One knows that the concept of minimality and maximality of left ideals and right ideals play an important role in semigroups. In this paper, we extend this concept to consider in n-ary semigroups. A number of results concerning relationships between minimality and maximality of n-ideals of n-ary semigroups and n-simple (0-n-simple) n-ary semigroups as well as some characterizations of minimality and maximality of n-ideals of n-ary semigroups are given.

Rough soft n-ary semigroups based on a novel congruence relation and corresponding decision making

Rough soft n-ary semigroups based on a novel congruence relation and corresponding decision making

A characterization of n-associative, monotone, idempotent functions on an interval that have neutral elements

We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala–Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral element. We also present a complete characterization of idempotent monotone n-associative functions on an interval that have neutral elements.

Roughness in n-ary semigroups based on fuzzy ideals

A novel congruence relation U(μ, t )o n ann-ary semigroup S is established. We show that U(μ, t) is a congruence relation on an n-ary semigroup S if μ is a fuzzy ideal of S. Based on this idea, we construct the lower and upper approximations in an n-ary semigroup. Furthermore, we introduce the notions of rough ideals and rough prime ideals by means of fuzzy ideals of an n-ary semigroup. In particular, the concepts of rough n-ary semigroups, rough homomorphisms are introduced and some relative properties are also investigated.

On absorption in semigroups and $n$-ary semigroups

The notion of absorption was developed a few years ago by Barto and Kozik and immediately found many applications, particularly in topics related to the constraint satisfaction problem. We investigate the behavior of absorption in semigroups and n-ary semigroups (that is, algebras with one n-ary associative operation). In the case of semigroups, we give a simple necessary and sufficient condition for a semigroup to be absorbed by its subsemigroup. We then proceed to n-ary semigroups, where we conjecture an analogue of this necessary and sufficient condition, and prove that the conjectured condition is indeed necessary and sufficient for B to absorb A (where A is an n-ary semigroup and B is its n-ary subsemigroup) in the following three cases: when A is commutative, when |A-B|=1 and when A is an idempotent ternary semigroup.

On the (n, m)- semirings derived polynomially from infinite semidomains

In the paper [Marichal, J.-L. and Mathonet, P., A description of n-ary semigroups polynomial-derived from integral domains, Semigroup Forum, 83 (2011), No. 2, 241–249] the authors provide a complete classification of the nsemigroups, defined by polynomial functions over infinite commutative integral domains with identity (i.e., the n-semigroup structures polynomial derived from integral domains). We remark that some results from that paper can be extended for the n-semigroups polynomial-derived defined over infinite semidomains with zerosumfree element. Furthermore, in this note we give a description of (n, m)- semiring structures defined by polynomial functions over such semidomains.

Aczélian n-ary semigroups

We show that the real continuous, symmetric, and cancellative n-ary semigroups are topologically order-isomorphic to additive real n-ary semigroups. The binary case (n=2) was originally proved by Acz\'el (1949); there symmetry was redundant.

A description of n-ary semigroups polynomial-derived from integral domains

We provide a complete classification of the n-ary semigroup structures defined by polynomial functions over infinite commutative integral domains with identity, thus generalizing Głazek and Gleichgewicht’s classification of the corresponding ternary semigroups.

A Generalization of n-Ary Algebraic Systems

One of the aims of this article is to extract, whenever possible, the common elements of several seemingly different types of algebraic hypersystems. In achieving this, one discovers general concepts, constructions, and results which not only generalize and unify the known special situations, thus leading to an economy of presentation, but, being at a higher level of abstraction, can also be applied to entirely new situations, yielding significant information and giving rise to new directions. We shall consider a class of algebraic hypersystems which represent a generalization of semigroups, hypersemigroups, and n-ary semigroups. This new class of hypersystems is called n-ary hypersemigroups and properties of such hypersemigroups are investigated. On an n-ary hypersemigroup, we describe the smallest equivalence relation β* whose quotient is an ordinary n-ary semigroup.

Approximations in n-ary algebraic systems

Algebraic systems have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders and error-correcting codes. The theory of rough sets has emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy, or incomplete information. This paper is devoted to the discussion of the relationship between algebraic systems, rough sets and fuzzy rough set models. We shall restrict ourselves to algebraic systems with one n-ary operation and we investigate some properties of approximations of n-ary semigroups. We introduce the notion of rough system in an n-ary semigroup. Fuzzy sets, a generalization of classical sets, are considered as mathematical tools to model the vagueness present in rough systems.

Idempotents in n-ary Semigroups

We select a new type of elements of n-ary (n ≥ 3) semigroups with an idempotent and investigate properties of ideals connected with these elements.

Idempotents in n-ary Semigroups

Idempotents in n-ary Semigroups