Implicative and Disjunctive Prevarieties: Algebraic aspects of Deduction Theorem

An (equational) implication/disjunction system for a class of algebrasis a set of quadruple equations defining implication/disjunction of equalities in algebras of the class. Then, a prevariety (viz.… Lue lisää An (equational) implication/disjunction system for a class of algebrasis a set of quadruple equations defining implication/disjunction of equalities in algebras of the class. Then, a prevariety (viz. , an implicational class), i. e. , an abstract hereditary multiplicative class of algebras issaid to be finitely] implicative/disjunctive, provided it is generated by a class with finite] implication/disjunction system. One of preliminary general results of the book is that a pre]variety is implicative/disjunctive iff it hasrestricted equationally definable principal relative] congruences/(congruence diagonal )meets (REDP R]C/ (CD)M) and isthe prevariety generated by its relatively] simple/finitely-subdirectly-irreducible membersiff both has REDP R]C/CDM and is relatively ]semi-simple/congruence-fmi-based. In particular, a quasi]variety is implicative/disjunctive iff itboth has REDP R]C and is relatively ]semi-simple/just has REDP R]CDM. And what is more, we prove that any class K of algebras ofa given algebraic signature S generates the quasivariety being a variety, whenever, for some subsinature S' of S, K-S' has a finite implication systemand generates the quasivariety being a variety. As for disjunctive pre]varieties, we also prove that these are relatively] congruence-distributive. This, in particular, implies the relative ]congruence-distributivity of (finitely )implicative quasi(pre)]varieties. And what is more, it collectively with Jonsson's Ultrafilter Lemma imply that any implicative quasivariety is a variety iff it is congruence-distributive and semi-simple. At last, we obtain congruence characterizations of finitely ]disjunctive/implicative (pre/quasi)varieties. In this connection, we also prove that there is no non-trivial implicative relatively congruence-Boolean prevariety. As a consequence, there is no non-trivial relatively] congruence-Boolean quasi]variety. In addition, we introduce the notion of semilattice congruence generalizing that ofideal one and prove that a quasi]variety has (R)EDP R]C iffit is relatively] (sub)directly semilattice iff it is relatively] (sub)directly ideal, and what is more, is relatively ](sub)directly filtral iff it both is relatively ]semi-simple and either has (R)EDP R]Cor is relatively] (sub)directly congruence-distributivewith (universally )axiomatizable class of relatively] simple(and trivial algebras) iff it is subdirecltly (non-)parmeterized implicative. As a consequence, a variety is discriminator iff it is arithmetical and semi-simple with universally axiomatizable class of simple and trivial algebras. And what is more, we prove that any prevariety generated by the algebra reductsof a finite class of finite prime filter expansions of latticeswith equality determinant is a finitely disjunctive quasivariety, the disjunction system being naturally defined by the equality determinant, with relative subdirectly-irreducibles, being exactly non-trivialalgebras embeddable into a member of the generating class, andis implicative iff it is relatively semi-simple, in which caseit is a variety iff it is semi-simple. And what is much more, we prove that any finite distributive latticeexpansion with a uniform equality determinant for all its primefilters has an implication system naturally defined by the equality determinant. These (merely, the former) prove to be well-applicable to both the varieties ofdistributive and De Morgan lattices( as well as Stone algebras)