Varieties of Monadic Heyting Algebras. Part III

This paper is the concluding part of [1] and [2], and it investigates the inner structure of the lattice (MHA) of all varieties of monadic Heyting algebras. For every n , we introduce and investigate varieties of depth n and cluster n, and present two partitions of (MHA), into varieties of depth n, and into varieties of cluster n. We pay a special attention to the lower part of (MHA) and investigate finite and critical varieties of monadic Heyting algebras in detail. In particular, we prove that there exist exactly thirteen critical varieties in (MHA) and that it is decidable whether a given variety of monadic Heyting algebras is finite or not. The representation of (MHA) is also given. All these provide us with a satisfactory insight into (MHA). Since (MHA) is dual to the lattice NExtMIPC of all normal extensions of the intuitionistic modal logic MIPC, we also obtain a clearer picture of the lattice structure of intuitionistic modal logics over MIPC.     

Varieties of Three-Valued Heyting Algebras with a Quantifier

This paper is devoted to the study of some subvarieties of the variety Qof Q-Heyting algebras, that is, Heyting algebras with a quantifier. In particular, a deeper investigation is carried out in the variety Q ₃ of three-valued Q-Heyting algebras to show that the structure of the lattice of subvarieties of Qis far more complicated that the lattice of subvarieties of Heyting algebras. We determine the simple and subdirectly irreducible algebras in Q ₃ and we construct the lattice of subvarieties (Q ₃ ) of the variety Q ₃ .     

On Extensions of Intermediate Logics by Strong Negation

In this paper we will study the properties of the least extension n() of a given intermediate logic by a strong negation. It is shown that the mapping from to n() is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n (). This summarizes results that can be found already in [13,14] and [4]. Furthermore, we determine the structure of the lattice of extensions of n(LC).     

Compatible Functions in Algebras Associated to Extensions of Positive Logic

In [4], Caicedo and Cignoli study compatible functions on Heytingalgebras and the corresponding logical properties of connectivesdefined on intuitionistic propositional calculus. In this paperwe study some aspects of compatible functions on the algebrasassociated to positive propositional calculus and successiveextensions of it: intuitionistic calculus itself, the modalsymmetric propositional calculus of Moisil and n-valued ukasiewiczpropositional calculus.     

On Complete Representations of Reducts of Polyadic Algebras

Following research initiated by Tarski, Craig and Németi, and futher pursued by Sain and others, we show that for certain subsets G of ^ω ω, atomic countable G polyadic algebras are completely representable. G polyadic algebras are obtained by restricting the similarity type and axiomatization of ω-dimensional polyadic algebras to finite quantifiers and substitutions in G. This contrasts the cases of cylindric and relation algebras. Presented by Robert Goldblatt     

From Semirings to Residuated Kleene Lattices

We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-*. An investigation of congruence properties (e-permutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Finally we define (one-sorted) residuated Kleene lattices with tests to complement two-sorted Kleene algebras with tests.     

Geometric analogue of holographic reduced representation

Holographic reduced representations (HRRs) are distributed representations of cognitive structures based on superpositions of convolution-bound n-tuples. Restricting HRRs to n-tuples consisting of ±1, one reinterprets the variable binding as a representation of the additive group of binary n-tuples with addition modulo 2. Since convolutions are not defined for vectors, the HRRs cannot be directly associated with geometric structures. Geometric analogues of HRRs are obtained if one considers a projective representation of the same group in the space of blades (geometric products of basis vectors) associated with an arbitrary n-dimensional Euclidean (or pseudo-Euclidean) space. Switching to matrix representations of Clifford algebras, one can always turn a geometric analogue of an HRR into a form of matrix distributed representation. In typical applications the resulting matrices are sparse, so that the matrix representation is less efficient than the representation directly employing the rules of geometric algebra. A yet more efficient procedure is based on ‘projected products’, a hierarchy of geometrically meaningful n-tuple multiplication rules obtained by combining geometric products with projections on relevant multivector subspaces. In terms of dimensionality the geometric analogues of HRRs are in between holographic and tensor-product representations.     

Subdirectly Irreducible Modal Algebras and Initial Frames

The duality between general frames and modal algebras allows to transfer a problem about the relational (Kripke) semantics into algebraic terms, and conversely. We here deal with the conjecture: the modal algebra A is subdirectly irreducible (s.i.) if and only if the dual frame A* is generated. We show that it is false in general, and that it becomes true under some mild assumptions, which include the finite case and the case of K4. We also prove that a Kripke frame F is generated if and only if the dual algebra F* is s.i. The technical result is that A is s.i. when the set of points which generate the dual frame A* is not of zero measure.