LOGlCS PROJECTIVELY GENERATED FROM [ A ] = (F4, [(l}]) BY , 4 SET OF HOMOMORPHISMS bg VENTURA VERDT? in Barcelona (Spain) ' ) 0. According to [2], an abstract logic of similarity type t is a pair (9, C) or (.Y, U) consisting of a finitary abstract algebra Y of type t together with a closure operator C on 45' or a closure system V on S, where S is the carrier of 9. I n this paper we study the abstract logic 2 ' = (.Y, V) of type (2) projectively generated from the abstract logic [JH] = (F4, [{l}]) by a set 9
This paper is divided in two parts. 111 the first one we obtain some consequences of the Deduction Property. Some of these consequences are a generalization in terms of closure operators and closure systems of several results of A. MONTEIRO [6] end A. DIEGO [4]. These results together with some properties of the projective and inductive generation of logics and some properties of the bi-logical morphisms are the basic tools used in the proofs of the theorems of the second part, where we obtain characterizations of the logics projectively generated from [A]. Theorem 3 of this paper is a generalization of theorem 3 of [ l ] in the following way: A4ccording to theorem 2 of [l], a classical logic is an abstract logic 9 = (9, C) such that Y = (8, ', v) is an algebra of type (1.2) and C satisfies (i) C' is algebraic (or finite), (ii) the Deduction Property relative to -+ (where x + y = 2' v y), (iii) Taut(Y) E C(0) (whereTaut(Y) = { z ~X : v ( x ) = 1 forallv~Hom(Y,({0,1},',v))}).
Theorem 3 of [l] states that 9 is classical iff there exists a bi-logical morphism from 9
to 9' = (.Y,, Vl), where Y', is a Boolean algebra and V, is the closure system of all filters of 9,. Our theorem 3 states that if a logic of type (2) satisfies the conditions (i), (ii) listed above and the condition (iii')
C ( 0 )
(which is a generalization of (iii)), then there exists a bi-logical morphism from 9 to zp1 = ( 4al, V,), where 9, is an implication algebra and is the closure system of all implicative filters of Y , , and conversely.
') The author is indebted to Dr. J. M. FONT for his helpfpl remarks. 2, See [a], definitions VI.1 and X.2. 3, This notation comes from POST'S classification of the two-element algebras. 4, See the definition of p. 237 of this paper. ') See [ 2 ] , definition 11.5.