Tomasz Kowalski

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.

On normal-valued basic pseudo-hoops

We show that every pseudo-hoop satisfies the Riesz decomposition property. We visualize basic pseudo-hoops by functions on a linearly ordered set. Finally, we study normal-valued basic pseudo-hoops giving a countable base of equations for them.

On decomposition of pseudo BL-algebras

We show that under some conditions, imposed on coatoms and maximal idempotents of a pseudo BL-algebra, we can decompose a pseudo BL-algebra M as an ordinal sum and we show that then M is linearly ordered. We investigate pseudo BL-algebras with a unique coatom a and with a maximal idempotent, and analyze two main situations: either an = an+1 holds for some n ≥ 1, or an > an+1 hold for any n ≥ 1. We note that there exist (subdirectly irreducible) algebras with two coatoms that are not linearly ordered, so the restriction to a single coatom is natural.

Analytic cut and interpolation for bi-intuitionistic logic

We prove that certain natural sequent systems for bi-intuitionistic logic have the analytic cut property. In the process we show that the (global) subformula property implies the (local) analytic cut property, thereby demonstrating their equivalence. Applying a version of Maehara technique modified in several ways, we prove that bi-intuitionistic logic enjoys the classical Craig interpolation property and Maximova variable separation property; its Hallden completeness follows.

Quasi-discriminator varieties

We generalize the notion of discriminator variety in such a way as to capture several varieties of algebras arising mainly from fuzzy logic. After investigating the extent to which this more general concept retains the basic properties of discriminator varieties, we give both an equational and a purely algebraic characterization of quasi-discriminator varieties. Finally, we completely describe the lattice of subvarieties of the pure pointed quasi-discriminator variety, providing an explicit equational base for each of its members.

Computable Isomorphisms of Boolean Algebras with Operators

In this paper we investigate computable isomorphisms of Boolean algebras with operators (BAOs). We prove that there are examples of polymodal Boolean algebras with finitely many computable isomorphism types. We provide an example of a polymodal BAO such that it has exactly one computable isomorphism type but whose expansions by a constant have more than one computable isomorphism type. We also prove a general result showing that BAOs are complete with respect to the degree spectra of structures, computable dimensions, expansions by constants, and the degree spectra of relations.

Two cooperative versions of the Guessing Secrets problem

We investigate two cooperative variants (with and without lies) of the Guessing Secrets problem, introduced in [L. Chung, R. Graham, F.T. Leighton, Guessing secrets, Electronic Journal of Combinatorics 8 (2001)] in the attempt to model an interactive situation arising in the World Wide Web, in relation to the efficient delivery of Internet content. After placing bounds on the cardinality of the smallest set of questions needed to win the game, we establish that the algebra of all the states of knowledge induced by any designated game is a pseudocomplemented lattice. In particular, its join semilattice reduct is embeddable into the corresponding reduct of the Boolean algebra 2^N^-^1, where N is the cardinality of the search space.

Self-implications in BCI

L. Humberstone asks whether every theorem of BCI provably implies φ → φ for some formula φ. R.K. Meyer conjectures that the axiom B does not imply any such “self-implication”. We prove a slightly stronger result, thereby confirming Meyer’s conjecture.

A Note on Monothetic BCI

In “Variations on a theme of Curry,” Humberstone conjectured that a certain logic, intermediate between BCI and BCK, is none other than monothetic BCI—the smallest extension of BCI in which all theorems are provably equivalent. In this note, we present a proof of this conjecture.

The Power of a Propositional Constant

Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of extensions of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4.