Abstract
AbstractFree algebras with an arbitrary number of free generators in varieties of BL-algebras generated by one BL-chain that is an ordinal sum of a finite MV-chain Ln, and a generalized BL-chain B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln, and free algebras in the variety of basic hoops generated by B. The Boolean products are taken over the Stone spaces of the Boolean subalgebras of idempotents of free algebras in the variety of MV-algebras generated by Ln.2000 Mathematics subject classification: primary 03G25, 03B50, 03B52, 03D35, 03G25, 08B20.
Highlights
Basic Fuzzy Logic (BL for short) was introduced by Hajek to formalize fuzzy logics in which the conjunction is interpreted by a continuous t-norm on the real segment [0, 1] and the implication by its corresponding adjoint
Important examples of subvarieties of BL-algebras are MV-algebras, linear Heyting algebras (that correspond to the superintuitionistic logic characterized by the axiom (P =» (2) v (G => P), see [25] for a historical account about this logic), PL-algebras
In [10] the first author described the finitely generated free algebras in the varieties of BL-algebras generated by a single BL-chain which is an ordinal sum of a finite MV-chain Ln and a generalized BL-chain B
Summary
Basic Fuzzy Logic (BL for short) was introduced by Hajek (see [19] and the references given there) to formalize fuzzy logics in which the conjunction is interpreted by a continuous t-norm on the real segment [0, 1] and the implication by its corresponding adjoint. In [10] the first author described the finitely generated free algebras in the varieties of BL-algebras generated by a single BL-chain which is an ordinal sum of a finite MV-chain Ln and a generalized BL-chain B The results of [10] were heavily based on the fact that the Boolean subalgebras of finitely generated algebras in the varieties generated by BLn-chains are finite. In the proof of this result a central role is played by the Moisil algebra reducts of algebras in MVn. Free algebras in varieties of BL-algebras generated by a single BLn-chain Ln l+J B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln and free algebras in the variety of basic hoops generated by B.
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