Minimal subvarieties of involutive residuated lattices with mingle axiom
نویسنده
چکیده
In this paper we study the number of minimal subvarieties of subvariety lattice. Since we can see that the subvariety lattice is dually isomorphic to the lattice of logics, the number of minimal subvarieties correspond to the number of maximal consistent logics. The maximal consistent logic over intuitionistic logic Int and FLew is only classical logic CL. For weaker logic number of minimal subvarieties may be uncountably many, see [1].
منابع مشابه
Minimal subvarieties of involutive residuated lattices
It is known that classical logic CL is the single maximal consistent logic over intuitionistic logic Int, which is moreover the single one even over the substructural logic FLew. On the other hand, if we consider maximal consistent logics over a weaker logic, there may be uncountablymany of them. Since the subvariety lattice of a given variety V of residuated lattices is dually isomorphic to th...
متن کاملA . M . Wille Minimal Varieties of Involutive Residuated Lattices
We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.
متن کاملAn approach to the variety IMTL(3)
We characterize, classify and axiomatize all subvarieties of IMTL(3), the variety of IMTL-algebras given by the equation ¬(x3) ∨ x ≈ >. We also give a canonical set of generators for each one. Introduction and preliminaries An IMTL-algebra is a bounded residuated lattice A = 〈A,∧,∨, ∗,→,¬,⊥,>〉, where ∧ and ∨ are meet and join, 〈∗,→〉 is a residuated pair, ¬ is the associated negation (i.e. ¬x =d...
متن کاملMinimal varieties of residuated lattices
In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of [12]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Fi...
متن کاملRelation Algebras As Expanded FL-Algebras
This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is t...
متن کامل