215 research outputs found
A Note on Direct Products, Subreducts and Subvarieties of PBZ*--lattices
PBZ*--lattices are bounded lattice--ordered structures arising in the study
of quantum logics, which include orthomodular lattices, as well as
antiortholattices. Antiortholattices turn out not only to be directly
irreducible, but also to have directly irreducible lattice reducts. Their
presence in varieties of PBZ*--lattices determines the lengths of the subposets
of dense elements of the members of those varieties. The variety they generate
includes two disjoint infinite ascending chains of subvarieties, and the
lattice of subvarieties of the variety of pseudo--Kleene algebras can be
embedded as a poset in the lattice of subvarieties of its subvariety formed of
its members that satisfy the Strong De Morgan condition. We obtain
axiomatizations for all members of a complete sublattice of the lattice of
subvarieties of this latter variety axiomatized by the Strong De Morgan
identity with respect to the variety generated by antiortholattices.Comment: 18 page
Almost structural completeness; an algebraic approach
A deductive system is structurally complete if its admissible inference rules
are derivable. For several important systems, like modal logic S5, failure of
structural completeness is caused only by the underivability of passive rules,
i.e. rules that can not be applied to theorems of the system. Neglecting
passive rules leads to the notion of almost structural completeness, that
means, derivablity of admissible non-passive rules. Almost structural
completeness for quasivarieties and varieties of general algebras is
investigated here by purely algebraic means. The results apply to all
algebraizable deductive systems.
Firstly, various characterizations of almost structurally complete
quasivarieties are presented. Two of them are general: expressed with finitely
presented algebras, and with subdirectly irreducible algebras. One is
restricted to quasivarieties with finite model property and equationally
definable principal relative congruences, where the condition is verifiable on
finite subdirectly irreducible algebras.
Secondly, examples of almost structurally complete varieties are provided
Particular emphasis is put on varieties of closure algebras, that are known to
constitute adequate semantics for normal extensions of S4 modal logic. A
certain infinite family of such almost structurally complete, but not
structurally complete, varieties is constructed. Every variety from this family
has a finitely presented unifiable algebra which does not embed into any free
algebra for this variety. Hence unification in it is not unitary. This shows
that almost structural completeness is strictly weaker than projective
unification for varieties of closure algebras
Expanding FLew with a Boolean connective
We expand FLew with a unary connective whose algebraic counterpart is the
operation that gives the greatest complemented element below a given argument.
We prove that the expanded logic is conservative and has the Finite Model
Property. We also prove that the corresponding expansion of the class of
residuated lattices is an equational class.Comment: 15 pages, 4 figures in Soft Computing, published online 23 July 201
Incidence combinatorics of resolutions
We introduce notions of combinatorial blowups, building sets, and nested sets
for arbitrary meet-semilattices. This gives a common abstract framework for the
incidence combinatorics occurring in the context of De Concini-Procesi models
of subspace arrangements and resolutions of singularities in toric varieties.
Our main theorem states that a sequence of combinatorial blowups, prescribed by
a building set in linear extension compatible order, gives the face poset of
the corresponding simplicial complex of nested sets. As applications we trace
the incidence combinatorics through every step of the De Concini-Procesi model
construction, and we introduce the notions of building sets and nested sets to
the context of toric varieties.
There are several other instances, such as models of stratified manifolds and
certain graded algebras associated with finite lattices, where our
combinatorial framework has been put to work; we present an outline in the end
of this paper.Comment: 20 pages; this is a revised version of our preprint dated Nov 2000
and May 2003; to appear in Selecta Mathematica (N.S.
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
State morphism MV-algebras
We present a complete characterization of subdirectly irreducible MV-algebras
with internal states (SMV-algebras). This allows us to classify subdirectly
irreducible state morphism MV-algebras (SMMV-algebras) and describe single
generators of the variety of SMMV-algebras, and show that we have a continuum
of varieties of SMMV-algebras
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