Journal of Algebraic Hyperstructures and Logical Algebras
Volume:5 Issue: 1, Winter 2024

Two-generated free MV -algebras are described in the variety of Sωn -algebras

MTL-algebras, the algebraic counterparts of Monoidal t-norm logic are studied and Chinese Remainder Theorem for MTL-algebras is established. Semilocal MTL– algebras are introduced and briefly characterized.

In this survey we consider the variety of unsharp orthomodular lattices. We will see that there are pretty smooth conditions that neatly generalize a great deal of the theory of orthomodular lattices. In particular, we will characterize the concept of block in this framework towards an appropriate notion of commutativity. Then, we capitalize on this fact and describe a categorical equivalence between orthomodular lattices, with fixed p-filters, and the variety of unsharp orthomodular lattices.

Some results about set theory and model theory of MV-algebras are collected, which allows us to count MV-algebras with various properties.

The concepts of SR-fuzzy quasi-subalgebras and (ultra) SR-fuzzy filters over Sheffer stroke BL-algebras are introduced for the study of quasi-subalgebras and (ultra) filters using the SR-fuzzy set, and various properties are investigated accordingly. The relationship between SR-fuzzy quasi-subalgebras and SR-fuzzy filters is considered, and the characterizations of an (ultra) SR-fuzzy filter are discussed.

L-algebras, connected to algebraic logic and quantum structures, were first introduced by Rump [13]. In [5], the number of some finite L-algebras is counted. But classifying and finding examples of L-algebras is not always an easy task and is very important. So, in this paper, we classify the set of all non-isomorphic L-algebras up to order 4. For this, we define the notion of partial and total conditions and by using them, we characterize all of the L-algebras of orders 2, 3 and 4. We prove that that there are 5 L-algebras of order 3 and 44 L-algebras of order 4, up to isomorphism.

This paper explores the construction of ordinal sums in the class of equality algebras. Although the ordinal sum of linear equality algebras remains linear, this property might not hold for representable equality algebras in general. We present (prime) deductive systems for the ordinal sum of each family of equality algebras, which help us find a necessary and sufficient condition under which an ordinal sum of equality algebras is representable.

Intermediate quantifiers are expressions of natural language, for example “most, almost all, many, a few” using which we quantify a number of some objects in a given universe. We have shown in [23] that all valid syllogisms with intermediate  quantifiers are a consequence of only two algebraic inequalities and one equality. The result was obtained in the formalism  of Lukasiewicz fuzzy type theory whose truth values form a linearly ordered complete MV-algebra. In this paper we will  prove that the same holds if we replace MV-algebra by a much more general IEQ-algebra (involutive EQ-algebra).