66 research outputs found
Order, algebra, and structure: lattice-ordered groups and beyond
This thesis describes and examines some remarkable relationships existing between seemingly quite different properties (algebraic, order-theoretic, and structural) of ordered groups. On the one hand, it revisits the foundational aspects of the structure theory of lattice-ordered groups, contributing a novel systematization of its relationship with the theory of orderable groups. One of the main contributions in this direction is a connection between validity in varieties of lattice-ordered groups, and orders on groups; a framework is also provided that allows for a systematic account of the relationship between orders and preorders on groups, and the structure theory of lattice-ordered groups. On the other hand, it branches off in new directions, probing the frontiers of several different areas of current research. More specifically, one of the main goals of this thesis is to suitably extend results that are proper to the theory of lattice-ordered groups to the realm of more general, related algebraic structures; namely, distributive lattice-ordered monoids and residuated lattices. The theory of lattice-ordered groups provides themain source of inspiration for this thesis’ contributions on these topics
Quasivarieties of Wajsberg hoops
In this paper we deal with quasivarieties of residuated structures which form the equivalent algebraic semantics of a positive frag-
ment of some substructural logic. Our focus is mainly on varieties and quasivarieties of Wajsberg hoops, which are the equivalent
algebraic semantics of the positive fragment of Łukasiewicz many-valued logic. In particular we study the lattice of subquasivari-
eties of Wajsberg hoops and we describe completely all the subvarieties of Wajsberg hoops that are primitive. Though the treatment
is mostly algebraic in nature, there are obvious connections with the underlying logic
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 of \brl s has GAP iff there is an unary term such
that satisfies the equations and
, for some }. Using this
characterization, we show that for any variety of bounded
residuated lattice satisfying GAP there is such that the equation
holds in , that is, . 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 such that it satisfies the
equation .} 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
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