151 research outputs found
Exploiting the Layout Engine to Assess Diagram Completions
A practicable approach to diagram completion is to first compute model completions on the abstract syntax level. These can be translated to corresponding diagram changes by the layout engine afterwards. Normally, several different model completions are possible though. One way to deal with this issue is to let the user choose among them explicitly, which is already helpful. However, such a choice step is a quite time-consuming interruption of the editing process. We argue that users often are mainly interested in completions that preserve their original diagram as far as possible. This criterion cannot be checked on the abstract syntax level though. In fact, minimal model changes might still result in enormous changes of the original diagram. Therefore, we suggest to use the layout engine in advance for assessing all possible model completions with respect to the diagram changes they eventually cause
Model-completion of varieties of co-Heyting algebras
It is known that exactly eight varieties of Heyting algebras have a
model-completion, but no concrete axiomatisation of these model-completions
were known by now except for the trivial variety (reduced to the one-point
algebra) and the variety of Boolean algebras. For each of the six remaining
varieties we introduce two axioms and show that 1) these axioms are satisfied
by all the algebras in the model-completion, and 2) all the algebras in this
variety satisfying these two axioms have a certain embedding property. For four
of these six varieties (those which are locally finite) this actually provides
a new proof of the existence of a model-completion, this time with an explicit
and finite axiomatisation.Comment: 28 page
Coherence in Modal Logic
A variety is said to be coherent if the finitely generated subalgebras of its
finitely presented members are also finitely presented. In a recent paper by
the authors it was shown that coherence forms a key ingredient of the uniform
deductive interpolation property for equational consequence in a variety, and a
general criterion was given for the failure of coherence (and hence uniform
deductive interpolation) in varieties of algebras with a term-definable
semilattice reduct. In this paper, a more general criterion is obtained and
used to prove the failure of coherence and uniform deductive interpolation for
a broad family of modal logics, including K, KT, K4, and S4
Uniform interpolation and coherence
A variety V is said to be coherent if any finitely generated subalgebra of a
finitely presented member of V is finitely presented. It is shown here that V
is coherent if and only if it satisfies a restricted form of uniform deductive
interpolation: that is, any compact congruence on a finitely generated free
algebra of V restricted to a free algebra over a subset of the generators is
again compact. A general criterion is obtained for establishing failures of
coherence, and hence also of uniform deductive interpolation. This criterion is
then used in conjunction with properties of canonical extensions to prove that
coherence and uniform deductive interpolation fail for certain varieties of
Boolean algebras with operators (in particular, algebras of modal logic K and
its standard non-transitive extensions), double-Heyting algebras, residuated
lattices, and lattices
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
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