2,879 research outputs found
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
MV-algebras freely generated by finite Kleene algebras
If V and W are varieties of algebras such that any V-algebra A has a reduct
U(A) in W, there is a forgetful functor U: V->W that acts by A |-> U(A) on
objects, and identically on homomorphisms. This functor U always has a left
adjoint F: W->V by general considerations. One calls F(B) the V-algebra freely
generated by the W-algebra B. Two problems arise naturally in this broad
setting. The description problem is to describe the structure of the V-algebra
F(B) as explicitly as possible in terms of the structure of the W-algebra B.
The recognition problem is to find conditions on the structure of a given
V-algebra A that are necessary and sufficient for the existence of a W-algebra
B such that F(B) is isomorphic to A. Building on and extending previous work on
MV-algebras freely generated by finite distributive lattices, in this paper we
provide solutions to the description and recognition problems in case V is the
variety of MV-algebras, W is the variety of Kleene algebras, and B is finitely
generated--equivalently, finite. The proofs rely heavily on the Davey-Werner
natural duality for Kleene algebras, on the representation of finitely
presented MV-algebras by compact rational polyhedra, and on the theory of bases
of MV-algebras.Comment: 27 pages, 8 figures. Submitted to Algebra Universali
Model completions for universal classes of algebras: necessary and sufficient conditions
Necessary and sufficient conditions are presented for the (first-order)
theory of a universal class of algebraic structures (algebras) to admit a model
completion, extending a characterization provided by Wheeler. For varieties of
algebras that have equationally definable principal congruences and the compact
intersection property, these conditions yield a more elegant characterization
obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski.
Moreover, it is shown that under certain further assumptions on congruence
lattices, the existence of a model completion implies that the variety has
equationally definable principal congruences. This result is then used to
provide necessary and sufficient conditions for the existence of a model
completion for theories of Hamiltonian varieties of pointed residuated
lattices, a broad family of varieties that includes lattice-ordered abelian
groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of
pointed residuated lattices admits a model completion, it must have
equationally definable principal congruences. In particular, the theories of
lattice-ordered abelian groups and MV-algebras do not have a model completion,
as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is
shown that certain varieties of pointed residuated lattices generated by their
linearly ordered members, including lattice-ordered abelian groups and
MV-algebras, can be extended with a binary operation in order to obtain
theories that do have a model completion.Comment: 32 page
On some Properties of Quasi MV Algebras and √quasi-MV Algebras. Part III.
In the present paper, which is a sequel to [14] and [3], we investigate further the structure theory of quasi-MV algebras and √′quasi-MV algebras. In particular: we provide an improved version of the subdirect representation theorem for both varieties; we characterise the Ursini ideals of quasi-MV algebras; we establish a restricted version of J´onsson’s lemma, again for both varieties; we simplify the proof of standard completeness for the variety of √′ quasi-MV algebras; we show that this same variety has the finite embeddability property; finally, we investigate the structure of the lattice of subvarieties of √′quasi-MV algebras
On some properties of quasi-MV algebras and square root quasi-MV algebras, IV
In the present paper, which is a sequel to
[20, 4, 12], we investigate further the structure theory of quasiMV
algebras and √0quasi-MV algebras. In particular: we provide
a new representation of arbitrary √0qMV algebras in terms
of √0qMV algebras arising out of their MV* term subreducts of
regular elements; we investigate in greater detail the structure
of the lattice of √0qMV varieties, proving that it is uncountable,
providing equational bases for some of its members, as well as
analysing a number of slices of special interest; we show that the
variety of √0qMV algebras has the amalgamation property; we
provide an axiomatisation of the 1-assertional logic of √0qMV
algebras; lastly, we reconsider the correspondence between Cartesian
√0qMV algebras and a category of Abelian lattice-ordered
groups with operators first addressed in [10]
A representation theorem for integral rigs and its applications to residuated lattices
We prove that every integral rig in Sets is (functorially) the rig of global
sections of a sheaf of really local integral rigs. We also show that this
representation result may be lifted to residuated integral rigs and then
restricted to varieties of these. In particular, as a corollary, we obtain a
representation theorem for pre-linear residuated join-semilattices in terms of
totally ordered fibers. The restriction of this result to the level of
MV-algebras coincides with the Dubuc-Poveda representation theorem.Comment: Manuscript submitted for publicatio
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