{ "item_title" : "Implicative and Disjunctive Prevarieties", "item_author" : [" Alexej P. Pynko "], "item_description" : "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 befinitely] implicative/disjunctive, provided it is generated by a class withfinite] implication/disjunction system.One of preliminary general results of the book is that apre]variety is implicative/disjunctive iff it hasrestricted equationally definable principalrelative] congruences/(congruence diagonal )meets (REDP R]C/ (CD)M) and isthe prevariety generated by itsrelatively] simple/finitely-subdirectly-irreducible membersiff both has REDP R]C/CDM and isrelatively ]semi-simple/congruence-fmi-based.In particular, aquasi]variety is implicative/disjunctive iff itboth has REDP R]C and isrelatively ]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 disjunctivepre]varieties, we also prove that these arerelatively] congruence-distributive.This, in particular, implies therelative ]congruence-distributivity of (finitely )implicativequasi(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 offinitely ]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-trivialrelatively] congruence-Booleanquasi]variety.In addition, we introduce the notion of semilattice congruence generalizing that ofideal one and prove that aquasi]variety has (R)EDP R]C iffit isrelatively] (sub)directly semilattice iff it isrelatively] (sub)directly ideal, and what is more, isrelatively ](sub)directly filtral iff it both isrelatively ]semi-simple and either has (R)EDP R]Cor isrelatively] (sub)directly congruence-distributivewith (universally )axiomatizable class ofrelatively] 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)", "item_img_path" : "https://covers4.booksamillion.com/covers/bam/1/69/095/555/1690955554_b.jpg", "price_data" : { "retail_price" : "200.00", "online_price" : "200.00", "our_price" : "200.00", "club_price" : "200.00", "savings_pct" : "0", "savings_amt" : "0.00", "club_savings_pct" : "0", "club_savings_amt" : "0.00", "discount_pct" : "10", "store_price" : "" } }

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