Abstract:The concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters. In this paper we introduce an axiomatization which shows how several resembling theorems that had been separately proved for various algebras of logic can be given unique proofs within this axiomatic framework. We thus recapture theorems already known in the literature, as well as new ones. As a by-product we introduce the class of pre-BCK algebras.
MSC:03G05, 03G20, 03G25, 06A12, 06B10, 06D20, 06D35, 06F35 The concept of a deductive system, introduced by Diego [18,19], is an algebraic counterpart of a theory closed under modus ponens. It has become an important tool in algebra of logic (Georgescu [21] emphasizes that the term algebraic logic should be reserved to the study of logic by algebraic means, while the study of algebras related to logic, mainly from an algebraic point of view, should be referred to as algebra of logic). The structures occurring in algebra of logic are usually endowed with an operation →, called implication, and the concept of a deductive system makes sense for them. If the structure is also a (semi)lattice, then its deductive systems are usually (semi)lattice filters, and that is why they are sometimes called implicative filters, while the converse may or may not hold. It turns out that whenever deductive systems do not coincide with filters, another concept has been invented which is similar to or stronger than a filter and is equivalent to the concept of a deductive system. Besides, the lattice of deductive systems or equivalents of it has been studied in more detail; see e. and for all the involved classes of algebras 1 . We aim at algebraic direct proofs, without recourse to the logics described by the algebras under consideration.Our aim described above amounts to a desideratum stated by a Romanian mathematician Dan Barbilian: prove a theorem under no more hypotheses than necessary. That is why we gradually introduce classes of algebras constructed ad hoc for "economically" proving the theorems that interest us and from which we immediately recapture the theorems known in the literature.The paper is organized as follows. Firstly (Section 1) we deal only with deductive systems, then (Sections 2, 3) we relate them to other types of subsets, known as filters: product filters, filters (more exactly, semilattice filters), order filters and strong filters. Section 2 compares deductive filters to product filters, while comparison with other types of filters is done in Section 3. The criterion for this division was that product filters require the existence of an extra binary operation , while in Section 3 the framework is the same as in Section 1, with the little exception of strong filters (dealt with in Proposition 3.3 and Corollary 3.4). The reason for which the comparison with product filters precedes the other comparisons is that the important Propositions 3.1a and 3.1b rely on Proposition 2.3.The concept of a deductive system requires ...
