Starting from quasi-Wajsberg algebras (which are generalizations of Wajsberg algebras), whose regular sets are Wajsberg algebras, we introduce a theory of quasi-algebras versus, in parallel, a theory of regular algebras. We introduce the quasi-RM, quasi-RML, quasi-BCI, (commutative, positive implicative, quasi-implicative, with product) quasi-BCK, quasi-Hilbert and quasi-Boolean algebras as gen...

Filters of a lattice, maximal filters (ultrafilters), operation to create a filter generating by an element or by a nonempty set of elements of the lattice are discussed. Besides, there are introduced implicative lattices such that for every two elements there is an element being pseudo-complement of them. Some facts concerning these concepts are presented too, i.e. for any proper filter there ...

Molodtsov [8] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to the theory of BCK-algebras. The notions of positive implicative soft ideals and positive implicative idealistic soft BCK-algebras are intro...

The concept of residuated relational systems ordered under a quasi-order relation was introduced in 2018 by S. Bonzio and I. Chajda as structure $\mathfrak{A} = \langle A, \cdot, \rightarrow, 1, \preccurlyeq \rangle$, where $(A,\cdot)$ is commutative semigroup with the identity $1$ top element this monoid $\preccurlyeq$. In 2020, author analyzed concepts filters type algebraic structures. addit...

In this paper, we introduce implicative quasi-semi-lattices as a generalization of semi-lattices. We give the definition and present related properties an quasi-semi-lattice. Moreover, show that there exists isomorphism between lattice filters quasi-semi-lattice filter congruences on

In this paper, we first introduce the new concept of completely semiprime fuzzy ideals of an ordered semigroup S, which is an extension of completely semiprime ideals of ordered semigroup S, and investigate some its related properties. Especially, we characterize an ordered semigroup that is a semilattice of simple ordered semigroups in terms of completely semiprime fuzzy ideals of ordered semi...

Considering a commutative unital quantale L as the truth value table and using the tool of L-generalized convergence structures of stratified L-filters, this paper introduces a kind of fuzzy upper topology, called fuzzy S-upper topology, on L-preordered sets. It is shown that every fuzzy join-preserving L-subset is open in this topology. When L is a complete Heyting algebra, for every completel...

In this paper, the concept of semiprimary fuzzy ideals of an ordered semigroup S is introduced. Some characterizations for an ordered semigroup S to be a semilattice of archimedean (rarchimedean, t-archimedean) ordered subsemigroups are given by some binary relations on S and the fuzzy radical of fuzzy ideals (fuzzy right ideals, fuzzy bi-ideals) of S. Furthermore, some characterizations for an...