N-ary Groups Research Articles

Finitely Generated Abelian n-Ary Groups

Finitely Generated Abelian n-Ary Groups

The Haar measure of a profinite n-ary group

We prove that every profinite [Formula: see text]-ary group [Formula: see text] has a unique Haar measure [Formula: see text] and further for every measurable subset [Formula: see text], we have [Formula: see text] where [Formula: see text] and [Formula: see text] are the normalized Haar measures of the profinite groups [Formula: see text] and the Post cover [Formula: see text], respectively.

Geometry induced by flocks

Using the vectors and symmetry of affine geometry induced by the ternary quasigroup satisfying the para-associative laws, we found the conditions under which such quasigroup becomes a ternary group. The obtained results also give a simple characterization of semiabelian n-ary groups.

Automorphisms of n-Ary Groups

Isotopies and autotopies of n-ary groups are described. As a consequence, we obtain various characterizations of the group of automorphisms of n-ary groups. We also determine the number of automorphisms of a given n-ary group.

Graded Medial n-Ary Algebras and Polyadic Tensor Categories

Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or ε-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with n−1 associators of the arity 2n−1 satisfying a n2+1-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.

Ternary SU(3)-group symmetry and its possible applications in hadron-quark substructure. Towards a new spinor-fermion structure

The questions on the existence of the three color quark symmetry and three quark-lepton generations could have the origin associated with the new exotic symmetries outside the Cartan-Killing-Lie algebras/groups. Our long-term search for these symmetries has been began with our Calabi-Yau space classification on the basis of the n-ary algebra for the reflexive projective numbers and led us to the expansion of the binary n = 2 complex and hyper complex numbers in the framework of the n-ary complex and hyper-complex numbers with n = 3, 4, … where we constructed new Abelian and non-Abelian symmetries. We have studied then norm-division properties of the Abelian nary complex numbers and have built the infinite chain of the Abelian groups U(n–1) = [U(1) × … × U(1)](n–1). We have developed the n-ary holomorphic (polymorphic) analysis on the n-ary complex space NC{n}, which led us to the generalization of the quadratic Laplace equations for the harmonic functions. The generalized Laplace equations for the n-ary harmonic functions give us the n-th order homogeneous differential equations which are invariant with respect to the Abelian n-ary groups U(n–1) and with some new spatial properties. Further consideration of the non-Abelian n-ary hyper-complex numbers opens the infinite series of the non-Abelian TnSU(n)-Lie groups(n=3,4,…) and its corresponding tnsu(n) algebras. One of the exceptional features of these symmetry groups is the appearance of some new n-dimensional spinors that could lead to an extension of the concept of the SU(2)-spin, to the appearance of n-dimensional quantum structures -exotic “n-spinor” matter(n = 3, 4, … - maarcrions). It is natural to assume that these new exotic “quantum spinor states” could be candidates for the pra-matter of the quark-charge leptons or/and for the dark matter. We will be also interested in the detection of the exotic quantum ’n-spinor” matter in the neutrino and hadron experiments.

On Identities and m-Neutral Sequences of n-Ary Groups

In this paper, we study such polyadic analog of an identity of a group as m-neutral sequence. In particular, we prove that all Post’s equivalence classes of the free covering group of any n-ary group [where \(n = k(m - 1) + 1\) and \(k\ge 1\)] defined by m-neutral sequences form the \((k + 1)\)-ary group, which is isomorphic to the n-ary subgroup of all identities of the n-ary group in the case when \(m = 2\).

On identities of an n-ary group

Known and new results on identities of an [Formula: see text]-ary group are studied in this paper.

Soft [formula omitted]-Ary Subgroups

Soft set theory plays a vital role in solving many complicated problems with inherited uncertainty. An n-ary algebraic systems is a generalization of algebraic structures and it is the most natural way for the further development, deeper understanding of their properties. In this paper, we apply soft set theory to an n-ary algebraic systems and introduce the notions of soft n-ary groups and soft n-ary subgroups. Further, some operations on soft sets are extended to the former. Finally, we provide the characterization of soft n-ary subgroups over an n-ary group (G,f) and study their related properties.

In the class of all n-ary groups semi-commutativity, commutativity and affinity are identical concepts

This paper proves that the class of all commutative n-ary groups and the class of all affine n-ary groups coincide with the class of all semiabelian n-ary groups.

The structure of finite abelian n-ary groups

AbstractThe structure of finite abelian n-ary groups is completely described: it is reduced to the direct product of primary abelian semicyclic n-ary groups up to isomorphism.

Combinatorial aspects of [formula omitted]-ary polygroups and [formula omitted]-ary color schemes

In this paper, the concepts of full conjugation relations and color schemes on n-ary polygroups are introduced. An n-ary color scheme on an n-ary polygroup is a generalization of a color scheme on a polygroup. We define the extension of two n-ary polygroups as a construction for building a “big” n-ary polygroup from two “small” ones. Moreover, we introduce the class of n-ary groups by a full conjugation relation on n-ary polygroups. In addition, we prove that the direct product of two n-ary color schemes and the class of n-ary groups by a full conjugation relation on n-ary polygroups are chromatic.

Fuzzy subgroups of n-ary subgroups

We introduce a notion of t-fuzzy n-ary subgroup in an n-ary group (G, f) and have studied their related properties.

Subgroups of semicyclic n-ary groups

We give a complete description of n-ary subgroups of finite and infinite (Abelian and non-Abelian) n-ary groups defined on cyclic groups.

Product structure of the fuzzy n-ary factor polygroup

Fuzzy n-ary hypergroups were introduced as suitable generalizations of fuzzy polygroups and a special case of fuzzy n-ary hyper groups. The aim of this paper is to introduce the notion of fuzzy n-ary factor polygroups of a polygroup. Based on the fuzzy n-ary factor group, we also study the product structures of the generating fuzzy factor n-ary groups. At the end of the paper, we prove the fundamental theorem of isomorphism of fuzzy n-ary groups.

Contemporary Engineering and the Metaphysics of Artefacts

In the last two decades, the philosophy of artefacts has grown into a substantial field. Various philosophers have proposed and discussed ways of determining the nature and identity conditions of artefacts; others have scrutinised possible definitions of artefact functions. Simultaneously, cognitive psychologists have gathered empirical evidence about how people identify and distinguish artefacts. Furthermore, these two discipli nary groups appear to be interested in each other's proposals and results (Margolis and Laurence 2007). A shared starting point in this literature is that artefacts and their functions deserve independent treatment. Most philosophers agree that artefact kinds are not natural kinds; most cognitive psychologists agree that people distinguish artefacts from natural objects. In consequent attempts to develop an appropriately autonomous account of artefacts, two intuitions have commanded virtually all attention. One is that artefacts are related, more intimately than natural objects, to intentional actions. Some authors stress that recognising intentions is important or even necessary in identifying artefacts; others offer conceptual or empirical reasons to reject this. The other intuition is that artefacts are identified and recognised by their functions. Some authors regard artefact functions as essential properties; others deny this. Studies into the relation between artefact functions and intentional actions combine these two intuitions. In this paper, we reconsider these intuitions in the light of another feature of existing philosophical work on artefacts, namely its minimalis tic view of the actions and agent roles involved in dealings with artefacts. The model implicit in most of the literature is that of an artisan, a single person who produces relatively simple artefacts, starting with a (perhaps personal) desire and ending with a finished product. This artisan model

Applications of interval valued t-norms ( t-conorms) to fuzzy n-ary sub-hypergroups

In this paper, we study a generalization of group, hypergroup and n-ary group. Firstly, we define interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T ( t-conorm S ). We give a necessary and sufficient condition for, an interval-valued fuzzy subset to be an interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T ( t-conorm S ). Secondly, using the notion of image (anti image) and inverse image of a homomorphism, some new properties of interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup are obtained with respect to infinitely ∨ -distributive t-norms T ( ∧ -distributive t-conorms S ). Also, we obtain some results of T -product ( S -product) of the interval-valued fuzzy subsets for infinitely ∨ -distributive t-norms T ( ∧ -distributive t-conorms S ). Lastly, we investigate some properties of interval-valued fuzzy subsets of the fundamental n-ary group with infinitely ∨ -distributive t-norms T ( ∧ -distributive t-conorms S ).

Around the Hosszú–Gluskin theorem for [formula omitted]-ary groups

We survey the results related to the important Hosszú–Gluskin Theorem on n-ary groups adding 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 also be used for investigation of Q -independent subsets of semiabelian n-ary groups for some special families Q of mappings.

Remarks to Głazek'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.

On the Schur theorem for n-ary groups

We prove n-ary analogs of the well-known Schur theorem on the finiteness of a commutator subgroup of a group whose center is of finite index.