Rigid and super rigid quasigroups
The paper deals with quasigroups having a trivial group of automorphisms and a trivial group of autotopisms. Examples of such quasigroups and methods of their verification are given.
math.GR↗
arXiv subjects
Wieslaw A. Dudek
Publications and source records attributed to Wieslaw A. Dudek.
On prolongations of quasigroups
We prove that any quasigroup admissing complete or quasicomplete mapping has a prolongation to a quasigroup having one element more.
math.GR↗
Fuzzy n-ary groups as a generalization of Rosenfeld's fuzzy groups
The notion of an $n$-ary group is a natural generalization of the notion of a group and has many applications in different branches. In this paper, the notion of (normal) fuzzy $n$-ary subgroup of an $n$-ary group is introduced and some related properties are investigated. Characterizations of fuzzy $n$-ary subgroups are given.
math.RA↗
Ultra LI-ideals in lattice implication algebras and MTL-algebras
A mistake concerning the ultra \textit{LI}-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an \textit{LI}-ideal to be an ultra \textit{LI}-ideal are given. Moreover, the notion of an \textit{LI}-ideal is extended to MTL-algebras, the notions of a (prime, ultra, obstinate, Boolean) \textit{LI}-ideal and an \textit{ILI}-ideal of an MTL-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in MTL-algebra: (1) prime proper \textit{LI}-ideal and Boolean \textit{LI}-ideal, (2) prime proper \textit{LI}-ideal and \textit{ILI}-ideal, (3) proper obstinate \textit{LI}-ideal, (4) ultra \textit{LI}-ideal.
math.LO↗
Special types of intuitionistic fuzzy left h-ideals of hemirings
Characteristic, normal and completely normal intuitionistic fuzzy left $h$-ideals of hemirings are described.
math.RA↗
Remarks to Glazek's results on n-ary groups
It is a survey of the results obtained by K. Glazek's and his co-workers. We restrict our attention to the problems of axiomatizations of n-ary groups, classes of n-ary groups, properties of skew elements and homomorphisms induced by skew elements, constructions of covering groups, classifications and representations of n-ary groups. Some new results are added too.
math.HO↗
Stationary subsets of functional Menger $\cap$-algebras of multiplace functions
A functional Menger $\cap$-algebra is a set of n-place functions containing n projections and closed under the so-called Menger's compositions of n-place functions and the set-theoretic intersection of functions. We give the abstract characterization for these subsets of functional Menger $\cap$-algebras which contain functions with fixed points.
math.GM↗
Stabilizers of functional Menger systems
A functional Menger system is a set of $n$-place functions containing $n$ projections and closed under the so-called Menger's composition of $n$-place functions. We give the abstract characterization for subsets of these functional systems which contain functions having one common fixed point.
math.GM↗
Representations of Menger $(2,n)$-semigroups by multiplace functions
Investigation of partial multiplace functions by algebraic methods plays an important role in modern mathematics were we consider various operations on sets of functions, which are naturally defined. The basic operation for $n$-place functions is an $(n+1)$-ary superposition $[ ]$, but there are some other naturally defined operations, which are also worth of consideration. In this paper we consider binary Mann's compositions $\op{1},...,\op{n}$ for partial $n$-place functions, which have many important applications for the study of binary and $n$-ary operations. We present methods of representations of such algebras by $n$-place functions and find an abstract characterization of the set of $n$-place functions closed with respect to the set-theoretic inclusion.
math.RA↗
Around the Hosszú-Gluskin theorem for $n$-ary groups
We survey results related to the important Hosszú-Gluskin Theorem on $n$-ary groups adding also several new results and comments. The aim of this paper is to write all such results in uniform and compressive forms. Therefore some proofs of new results are only sketched or omitted if their completing seems to be not too difficult for readers. In particular, we show as the Hosszú-Gluskin Theorem can be used for evaluation how many different $n$-ary groups (up to isomorphism) exist on some small sets. Moreover, we sketch as the mentioned theorem can be also used for investigation of $\mathcal{Q}$-independent subsets of semiabelian $n$-ary groups for some special families $\mathcal{Q}$ of mappings.
math.GR↗
On intuitionistic fuzzy sub-hyperquasigroups of hyperquasigroups
The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we consider the intuitionistic fuzzification of the concept of sub-hyperquasigroups in a hyperquasigroup and investigate some properties of such sub-hyperquasigroups. In particular, we investigate some natural equivalence relations on the set of all intuitionistic fuzzy sub-hyperquasigroups of a hyperquasigroup.
math.GM↗
Representations of $(2,n)$-semigroups by multiplace functions
We describe the representations of $(2,n)$-semigroups, i.e. groupoids with $n$ binary associative operations, by partial $n$-place functions and prove that any such representation is a union of some family of representations induced by Schein's determining pairs.
math.RA↗
Basic concepts of ternary Hopf algebras
The theory of ternary semigroups, groups and algebras is reformulated in the abstract arrow language. Then using the reversing arrow ansatz we define ternary comultiplication, bialgebras and Hopf algebras and investigate their properties. The main property "to be binary derived" is considered in detail. The co-analog of Post theorem is formulated. It is shown that there exist 3 types of ternary coassociativity, 3 types of ternary counits and 2 types of ternary antipodes. Some examples are also presented.
math.QA↗
Bi-element representations of ternary groups
General properties of ternary semigroups and groups are considered. The bi-element representation theory in which every representation matrix corresponds to a pair of elements is built, connection with the standard theory is considered and several concrete examples are constructed. For clarity the shortened versions of classical Gluskin-Hosszú and Post theorems are given for them.
math.GR↗