104 research outputs found
Varieties of unary-determined distributive -magmas and bunched implication algebras
A distributive lattice-ordered magma (-magma)
is a distributive lattice with a binary operation that preserves joins
in both arguments, and when is associative then is an
idempotent semiring. A -magma with a top is unary-determined if
. These
algebras are term-equivalent to a subvariety of distributive lattices with
and two join-preserving unary operations . We obtain simple
conditions on such that is
associative, commutative, idempotent and/or has an identity element.
This generalizes previous results on the structure of doubly idempotent
semirings and, in the case when the distributive lattice is a Heyting algebra,
it provides structural insight into unary-determined algebraic models of
bunched implication logic. We also provide Kripke semantics for the algebras
under consideration, which leads to more efficient algorithms for constructing
finite models. We find all subdirectly irreducible algebras up to cardinality
eight in which is a closure operator, as well as all finite
unary-determined bunched implication chains and map out the poset of
join-irreducible varieties generated by them
Varieties of unary-determined distributive -magmas and bunched implication algebras
A distributive lattice-ordered magma (-magma)
is a distributive lattice with a binary operation that preserves joins
in both arguments, and when is associative then is an
idempotent semiring. A -magma with a top is unary-determined if
. These
algebras are term-equivalent to a subvariety of distributive lattices with
and two join-preserving unary operations . We
obtain simple conditions on such that is associative, commutative,
idempotent and/or has an identity element.
This generalizes previous results on the structure of doubly idempotent
semirings and, in the case when the distributive lattice is a Heyting algebra,
it provides structural insight into unary-determined algebraic models of
bunched implication logic. We also provide Kripke semantics for the algebras
under consideration, which leads to more efficient algorithms for constructing
finite models. We find all subdirectly irreducible algebras up to cardinality
eight in which is a closure operator, as well as all
finite unary-determined bunched implication chains and map out the poset of
join-irreducible varieties generated by them
On the Mereological Structure of Complex States of Affairs
The aim of this paper is to elucidate the mereological structure of complex states of affairs without relying on the problematic notion of structural universals. For this task tools from graph theory, lattice theory, and the theory of relational systems are employed. Our starting point is the mereology of similarity structures. Since similarity structures are structured sets, their mereology can be considered as a generalization of the mereology of sets.
The Logic of Partitions: Introduction to the Dual of the Logic of Subsets
Modern categorical logic as well as the Kripke and topological models of
intuitionistic logic suggest that the interpretation of ordinary
"propositional" logic should in general be the logic of subsets of a given
universe set. Partitions on a set are dual to subsets of a set in the sense of
the category-theoretic duality of epimorphisms and monomorphisms--which is
reflected in the duality between quotient objects and subobjects throughout
algebra. If "propositional" logic is thus seen as the logic of subsets of a
universe set, then the question naturally arises of a dual logic of partitions
on a universe set. This paper is an introduction to that logic of partitions
dual to classical subset logic. The paper goes from basic concepts up through
the correctness and completeness theorems for a tableau system of partition
logic
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