1,720,981 research outputs found
On Normal Forms in Lukasiewicz's Logic
No full textFormulas of n variables of Lukasiewicz sentential calculus can be represented, via McNaughton's theorem, by piecewise linear functions, with integer coefficients, from hypercube [0,1]n to [0,1], called McNaughton functions. As a consequence of McNaughton representation, a canonical form of a formula is obtained. Indeed, up to logical equivalence, any formula can be written as an infimum of finite suprema of formulas associated to McNaughton functions which are truncated functions to [0,1] of the restriction to [0,1]n of single hyperplanes, for short, called simple McNaughton functions. In the present paper the authors concern with the problem of presenting formulas of Lukasiewicz sentential calculus in normal from. The main results are: a) an axiomatic description of some classes of formulas having the property to be canonically mapped one-to-one onto the class of simple Mc Naughton functions; b) a normal form for Lukasiewicz sentential calculus, making use of formulas defined in (a); c) the polynomial complexity of formulas, in normal form, coming from a certain class described as in (a) is proved; d) the results described in (a), (b) and (c) are extended to Rational Lukasiewicz logic
Relative MV-algebras and relative homomorphisms
No full textIn the paper is defined the notion of relative subalgebra of an MV-algebra A. A particular case of this notion is the notion of an interval subalgebra of A, which has been already studied in the literature. Applying these notions, two new categories denoted as rMV and intMV are introduced. In both cases the objects are MV-algebras, but the homomorphisms are defined by means of relative subalgebras or by interval subalgebras, respectively. The relations occurring between these categories and the category of all MV-algebras with usual homomorphisms are investigated. The main results of the paper deal with one-generated free MV-algebras in the variety generated by the finite MV-chains Si , 0<i< p +1 (p varying over the set of all positive integers) and their relations to certain relative subalgebras of the cyclic free MV-algebra
Equational characterization of all varieties of MV-algebras
No full textAbstractIt is known that all subvarieties of MV-algebras are finitely axiomatizable. In the literature, one can find equational characterizations of certain subvarieties, such as MVn-algebras. In this paper we write down equational bases for all MV-varieties and prove a representation theorem for each subvariety
Boolean dominated MV-algebras
No full textIn this paper the authors study an obvious generalization of the hyperarchimedian MV-algebras: boolean dominated MV-algebras. Particularly they point out the wide difference between the class of the hyperarchimedian MV-algebras and the class of the Boolean dominated MV-algebras
Finiteness based results in BL-algebras
No full textBL-algebras were introduced by P. Hajek as algebraic structures of Basic Logic. The aim of the paper is a survey of known results about the structure of finite BL-algebras and natural dualities for varieties of BL-algebras. Extending the notion of ordinal sum of BL-algebras, a class of finite BL-algebras, actually BL-comets, which can be seen as a generalization of finite BL-chain, is characterized. Then, just using BL-comets, any finite BL-algebra can be represented as a direct product of BL-comets. This result can be seen as a generalization of the representation of finite MV-algebras as a direct product of finite MV-chains. Then it is shown the existence of a strong duality for each variety generated by one finite non-trivial totally ordered BL-algebra. As an application of the dualities, the injective and the weak injective members of these classes are described
Representations of Monadic MV-algebras
No full textRepresentations of monadic MV-algebra, the characterization of locally finite monadic MV-algebras, with axiomatization of them, definability of non-trivial monadic operators on finitely generated free MV-algebras are given. Moreover it is shown that a finitely generated m-relatively complete subalgebra of a finitely generated free MV-algebra is projective
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