The General Apple Property and Boolean terms in Integral Bounded Residuated Lattice-ordered Commutative Monoids

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 \emph{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: \emph{A variety $\mathsf{V}$ of \brl s has GAP iff there is an unary term $b(x)$ such that $\mathsf{V}$ satisfies the equations $b(x)\lor\neg b(x)\approx \top$ and $(x^k\to b(x))\cdot(b(x)\to k.x)\approx \top$, for some $k>0$}. Using this characterization, we show that for any variety $\mathsf{V}$ of bounded residuated lattice satisfying GAP there is $k>0$ such that the equation $k.x\lor k.\neg x\approx \top$ holds in $\mathsf{V}$, that is, $\mathsf{V} \subseteq \mathsf{WL_\mathsf{k}}$. As a consequence we improve Theorem 5.7 of \cite{CT12}, showing in theorem that a\emph{ variety of \brls\ has Boolean retraction term if and only if there is $k>0$ such that it satisfies the equation $k.x^k\lor k.(\neg k)^n\approx\top$.} 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 \cite{GLM12}). 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.Comment: 25 pages, 1 figure, 2 table

Projectivity in (bounded) integral residuated lattices

In this paper we study projective algebras in varieties of (bounded) commutative integral residuated lattices from an algebraic (as opposed to categorical) point of view. In particular we use a well-established construction in residuated lattices: the ordinal sum. Its interaction with divisibility makes our results have a better scope in varieties of divisibile commutative integral residuated lattices, and it allows us to show that many such varieties have the property that every finitely presented algebra is projective. In particular, we obtain results on (Stonean) Heyting algebras, certain varieties of hoops, and product algebras. Moreover, we study varieties with a Boolean retraction term, showing for instance that in a variety with a Boolean retraction term all finite Boolean algebras are projective. Finally, we connect our results with the theory of Unification

Representations and Completions for Ordered Algebraic Structures

The primary concerns of this thesis are completions and representations for various classes of poset expansion, and a recurring theme will be that of axiomatizability. By a representation we mean something similar to the Stone representation whereby a Boolean algebra can be homomorphically embedded into a field of sets. So, in general we are interested in order embedding posets into fields of sets in such a way that existing meets and joins are interpreted naturally as set theoretic intersections and unions respectively. Our contributions in this area are an investigation into the ostensibly second order property of whether a poset can be order embedded into a field of sets in such a way that arbitrary meets and/or joins are interpreted as set theoretic intersections and/or unions respectively. Among other things we show that unlike Boolean algebras, which have such a ‘complete’ representation if and only if they are atomic, the classes of bounded, distributive lattices and posets with complete representations have no first order axiomatizations (though they are pseudoelementary). We also show that the class of posets with representations preserving arbitrary joins is pseudoelementary but not elementary (a dual result also holds). We discuss various completions relating to the canonical extension, whose classical construction is related to the Stone representation. We claim some new results on the structure of classes of poset meet-completions which preserve particular sets of meets, in particular that they form a weakly upper semimodular lattice. We make explicit the construction of \Delta_{1}-completions using a two stage process involving meet- and join-completions. Linking our twin topics we discuss canonicity for the representation classes we deal with, and by building representations using a meet-completion construction as a base we show that the class of representable ordered domain algebras is finitely axiomatizable. Our method has the advantage of representing finite algebras over finite bases

The Reticulation of a Universal Algebra

The reticulation of an algebra $A$ is a bounded distributive lattice ${\cal L}(A)$ whose prime spectrum of filters or ideals is homeomorphic to the prime spectrum of congruences of $A$, endowed with the Stone topologies. We have obtained a construction for the reticulation of any algebra $A$ from a semi-degenerate congruence-modular variety ${\cal C}$ in the case when the commutator of $A$, applied to compact congruences of $A$, produces compact congruences, in particular when ${\cal C}$ has principal commutators; furthermore, it turns out that weaker conditions than the fact that $A$ belongs to a congruence-modular variety are sufficient for $A$ to have a reticulation. This construction generalizes the reticulation of a commutative unitary ring, as well as that of a residuated lattice, which in turn generalizes the reticulation of a BL-algebra and that of an MV-algebra. The purpose of constructing the reticulation for the algebras from ${\cal C}$ is that of transferring algebraic and topological properties between the variety of bounded distributive lattices and ${\cal C}$, and a reticulation functor is particularily useful for this transfer. We have defined and studied a reticulation functor for our construction of the reticulation in this context of universal algebra.Comment: 29 page