Arities and Aritizabilities of First-Order Theories

We study and describe possibilities for arities of elementary theories and of their expansions. Links for arities with respect to Boolean algebras, to disjoint unions and to compositions of structures are shown. Arities and aritizabilities are semantically characterized. The dynamics for arities of theories is described.

Sheffer operation in relational systems

The concept of a Sheffer operation known for Boolean algebras and orthomodular lattices is extended to arbitrary directed relational systems with involution. It is proved that to every such relational system, there can be assigned a Sheffer groupoid and also, conversely, every Sheffer groupoid induces a directed relational system with involution. Hence, investigations of these relational systems can be transformed to the treatment of special groupoids which form a variety of algebras. If the Sheffer operation is also commutative, then the induced binary relation is antisymmetric. Moreover, commutative Sheffer groupoids form a congruence distributive variety. We characterize symmetry, antisymmetry and transitivity of binary relations by identities and quasi-identities satisfied by an assigned Sheffer operation. The concepts of twist products of relational systems and of Kleene relational systems are introduced. We prove that every directed relational system can be embedded into a directed relational system with involution via the twist product construction. If the relation in question is even transitive, then the directed relational system can be embedded into a Kleene relational system. Any Sheffer operation assigned to a directed relational system $${\mathbf {A}}$$ A with involution induces a Sheffer operation assigned to the twist product of $${\mathbf {A}}$$ A .

Classical Mereology

The aim of this chapter is to provide a comprehensive introduction to classical mereology. It examines this theory by providing a clear and perspicuousaxiom system that isolates several important elements of any mereological theory. The chapter examines, algebraic, and set-theoretic models of classical mereology, sketching proofs of their equivalence. The new axiom system facilitates algebraic comparisons, showing that models of these axioms are complete Boolean algebras without a bottom element. Then set-theoretic models are presented, and are shown to satisfy the axioms. The chapter explains the important relationship between models and powersets, and the role of Stone’s Representation Theorem in this connection. Finally, a number of significant rival axiom systems using different mereological primitives are introduced.

On Boolean posets of numerical events

With many physical processes in which quantum mechanical phenomena can occur, it is essential to take into account a decision mechanism based on measurement data. This can be achieved by means of so-called numerical events, which are specified as follows: Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state $$s\in S$$ s ∈ S . A function $$p:S\rightarrow [0,1]$$ p : S → [ 0 , 1 ] is called a numerical event or alternatively, an S-probability. If a set P of S-probabilities is ordered by the order of real functions, it becomes a poset which can be considered as a quantum logic. In case the logic P is a Boolean algebra, this will indicate that the underlying physical system is a classical one. The goal of this paper is to study sets of S-probabilities which are not far from being Boolean algebras by means of the addition and comparison of functions that occur in these sets. In particular, certain classes of so-called Boolean posets of S-probabilities are characterized and related to each other and descriptions based on sets of states are derived.

Tense Operators on BL-algebras and its Applications

In this paper, the notions of tense operators and tense filters in \(BL\)-algebras are introduced and several characterizations of them are obtained. Also, the relation among tense \(BL\)-algebras, tense \(MV\)-algebras and tense Boolean algebras are investigated. Moreover, it is shown that the set of all tense filters of a \(BL\)-algebra is complete sublattice of \(F(L)\) of all filters of \(BL\)-algebra \(L\). Also, maximal tense filters and simple tense \(BL\)-algebras and the relation between them are studied. Finally, the notions of tense congruence relations in tense \(BL\)-algebras and strict tense \(BL\)-algebras are introduced and an one-to-one correspondence between tense filters and tense congruences relations induced by tense filters are provided.

A note on 3×3-valued Łukasiewicz algebras with negation

In 2004, C. Sanza, with the purpose of legitimizing the study of $$n\times m$$-valued Łukasiewicz algebras with negation (or $$NS_{n\times m}$$-algebras) introduced $$3\times 3$$-valued Łukasiewicz algebras with negation. Despite the various results obtained about $$NS_{n\times m}$$-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated $$NS_{3 \times 3}$$-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice $$\Lambda$$(NS$$_{3\times 3}$$) of all subvarieties of NS$$_{3\times 3}$$ and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of NS_$${3\times 3}$$.