Lattice-theoretic properties of algebras of logic
نویسندگان
چکیده
In the theory of lattice-ordered groups, there are interesting examples of properties — such as projectability — that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semilinear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a, 1] is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety.
منابع مشابه
Amalgamation and Interpolation in the Category of Heyting Algebras
This is the first of two papers describing how properties of open continuous maps between locales (which are the lattice-theoretic generalisation of topological spaces) can be used to give very straight-forward, constructive proofs of certain properties of first-order intuitionistic theories. The properties we have in mind are those of stability of a conservative interpretation of theories unde...
متن کاملLattice of full soft Lie algebra
In this paper, we study the relation between the soft sets and soft Lie algebras with the lattice theory. We introduce the concepts of the lattice of soft sets, full soft sets and soft Lie algebras and next, we verify some properties of them. We prove that the lattice of the soft sets on a fixed parameter set is isomorphic to the power set of a ...
متن کاملPreface: In memory of Wim Blok
algebraic logic: Full models, Frege systems, and metalogical properties he formulates an institutional analogue of the property of congruence and analyses how it helps in the preservation of other metalogical properties such as conjunction, disjunction, the deduction-detachment theorem, and two versions of reductio ad absurdum. In partial contrast, Raftery’s paper The equational definability of...
متن کاملAN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC
In this paper we extend the notion of degrees of membership and non-membership of intuitionistic fuzzy sets to lattices and introduce a residuated lattice with appropriate operations to serve as semantics of intuitionistic fuzzy logic. It would be a step forward to find an algebraic counterpart for intuitionistic fuzzy logic. We give the main properties of the operations defined and prove som...
متن کامل