Constructing denumerable matrices strongly adequate for pre-finite logics

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In [12] it is proved that no denumerable matrix is strongly adequate for the intui tionistic propositional logic (INT). Thus, the question naturally arises: what inter? mediate logics have denumerable matrices strongly adequate? It is obvious that every finite intermediate logic has a finite strongly adequate matrix. Following Maximova [8] let us call an intermediate logic pre-finite if it is a maximal one in the set of non-finite intermediate logics. In this paper each pre-finite intermediate logic is shown to have a denumerable matrix strongly adequate. Let 2B = , ~~1) be the free algebra in the class of all algebras of the similarity type free-generated by a denumerably infinite set F? W. The elements of Ware called words and denoted by a, ?,..., the elements of Vare called variables and denoted by x, y,..., the operations : a , v, ->, ~~| are called connectives (conjunction, disjunction, implication and negation connective respecti? vely). The familiar abbreviation a ?)A(?->a). Endomorphisms of the algebra 2B are called substitutions. The symbol Sb denotes the consequence operation in IF determined by the substitutions (for every X ? W, Sb(x) = \^J (e(X): e is a substitution)). The symbol Cn denotes the consequence operation in ^determi? ned by the theorems of INT and the detachment rule. If A ? Wand A = Cn(Sb(A)) 7^ W then we say that A is an intermediate logic. The consequence operation of an intermediate logic A is denoted by CnA (for every X ? W, CnA(X) = C?(lu A)). Any pair > where SD? is an algebra similar to 2B and D is a subset of the domain of SER is called a matrix. The operations of 9JI corresponding to the connectives are denoted by a^, v^, -^r, and "Ian respectively. Homomorphisms of 3B into ?ft are called StR-valuations. The symbol C > denotes the consequence ope? ration of the matrix (for every X ^ W and aeW, ae C (X) iff for every SD?-valuation v, v(X) ? D implies v(a) eD). To simplify the notations we will use the symbol E( (0) for denoting the content of the matrix > is said to be adequate (strongly adequate) for an interme? diate logic A iff E(yil, D) = A (C = CnA). An intermediate logic is called finite (pre-finite) iff it has a finite matrix adequate (it is maximal in the set of all intermediate logics having no finite matrix adequate). For investigations into intermediate logics the most important are matrices resul? ting from s.c. pseudo-Boolean algebras (see [10]). Let the symbol K denotes the class of all pseudo-Boolean algebras. The German capitals : 51, 93, ... are used to denote algebras of K and the corresponding Latin capitals : A, B, ... their domains. The symbols: 1^, 0^,