We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context.
The contribution is devoted to the question of the interchange of the construction of a quasiorder hypergroup from a quasiordered set and the factorization.
The main aim of this paper is to generalize the concept of vector space by the hyperstructure. We generalize some definitions such as hypersubspaces, linear combination, Hamel basis, linearly dependence and linearly independence. A few important results like deletion theorem, extension theorem, dimension theorem have been established in this hypervector space.
In this paper, we define and study the hyper S-posets over an ordered semihypergroup in detail. We introduce the hyper version of a pseudoorder in a hyper S-poset, and give some related properties. In particular, we characterize the structure of factor hyper S-posets by pseudoorders. Furthermore, we introduce the concepts of order-congruences and strong order-congruences on a hyper S-poset A, and obtain the relationship between strong order-congruences and pseudoorders on A. We also characterize...
Quasigroups were originally described combinatorially, in terms of existence and uniqueness conditions on the solutions to certain equations. Evans introduced a universal-algebraic characterization, as algebras with three binary operations satisfying four identities. Now, quasigroups are redefined as heterogeneous algebras, satisfying just two conditions respectively known as hypercommutativity and hypercancellativity.
AMS Subj. Classification: 03C05, 08B20Free algebras are very important in studying classes of algebras, especially varieties of algebras. Any algebra that belongs to a given variety of algebras can be characterized as a homomorphic image of a free algebra of that variety. Describing free algebras is an important task that can be quite complicated, since there is no general method to resolve this problem. The aim of this work is to investigate classes of groupoids, i.e. algebras with one binary operation,...
We investigate the structure and properties of -sub-semihypergroups, where is an arbitrary triangular norm on a given complete lattice . We study its structure under the direct product and with respect to the fundamental relation. In particular, we consider and , and investigate the connection between -sub-semihypergroups and the probability space.