2023/07/06 by Antoni Torrell, Torrell, Antoni Torrens
Computer Science · Mathematics · #03C05 #03G99 #06D35 #08A72 #Advanced Algebra and Logic #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2307.02944
openalex publication_date 2023/07/06 · openalex created_date 2023/07/08 · openalex updated_date 2026/07/28
In this paper we give equational presentations of the varieties of \em integral bounded residuated lattice-ordered commutative monoids (bounded residuated lattices for short) satisfying the General Apple Property (GAP), that is, varieties in which all of its directly indecomposable members are local. This characterization is given by means of Boolean terms: \emphA variety V of \brl s has GAP iff there is an unary term b(x) such that V satisfies the equations b(x)∨¬ b(x)≈ \top and (xk→ b(x))⋅(b(x)→ k.x)≈ \top, for some k>0. Using this characterization, we show that for any variety V of bounded residuated lattice satisfying GAP there is k>0 such that the equation k.x∨ k.¬ x≈ \top holds in V, that is, V ⊆ WLk. As a consequence we improve Theorem 5.7 of \citeCT12, showing in theorem that a variety of \brls has Boolean retraction term if and only if there is k>0 such that it satisfies the equation k.xk∨ k.(¬ x)k≈\top. We also see that in Bounded residuated lattices GAP is equivalent to Boolean lifting property (BLP) and so, it is equivalent to quasi-local property (in the sense of \citeGLM12). Finally, we prove that a variety of \brl s has GAP and its semisimple members form a variety if and only if there exists an unary term which is simultaneously Boolean and radical for this variety.