Tomasz Stanisław Kowalski

Jagiellonian University | UJ · Department of Logic

We investigate a semigroup construction generalising the two-sided wreath product. We develop the foundations of this construction and show that for groups it is isomorphic to the usual wreath product. We also show that it gives a slightly finer version of the decomposition in the Krohn–Rhodes Theorem, in which the three-element flip-flop monoid is...

In the the present contribution, we prove an Omitting Types Theorem (OTT) for an arbitrary fragment of hybriddynamic first-order logic with rigid symbols (i.e. symbols with fixed interpretations across worlds) closed under negation and retrieve. The logical framework can be regarded as a parameter and it is instantiated by some well-known hybrid an...

Conventional Ramsey-theoretic investigations for edge-colourings of complete graphs are framed around avoidance of certain configurations. Motivated by considerations arising in the field of Qualitative Reasoning, we explore edge colourings that in addition to forbidding certain triangle configurations also require others to be present. These condi...

Lindström’s theorem characterizes first-order logic in terms of its essential model theoretic properties. One cannot gain expressive power extending first-order logic without losing at least one of compactness or downward Löwenheim–Skolem property. We cast this result in an abstract framework of institution theory, which does not assume any interna...

It is known that in the lattice of normal extensions of the logic KTB there are unique logics of codimensions 1 and 2, namely, the logic of a single reflexive point, and the logic of the total relation on two points. A natural question arises about the cardinality of the set of normal extensions of KTB of codimension 3. Generalising two finite exam...

It is well known that the subvariety lattice of the variety of relation algebras has exactly three atoms. The (join-irreducible) covers of two of these atoms are known, but a complete classification of the (join-irreducible) covers of the remaining atom has not yet been found. These statements are also true of a related subvariety lattice, namely t...

We study splittings or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the non-splitting lemma, which when combined with some variety-specific constructions, yields each of our negative results: the variety of commutative integral residuated lattices contains no splitting algebras, and in the v...

We generalize the characterization of elementary equivalence by Ehrenfeucht-Fraïssé games to arbitrary institutions whose sentences are finitary. These include many-sorted first-order logic, higher-order logic with types, as well as a number of other logics arising in connection to specification languages. The gain for the classical case is that th...

A sequent system is used to give alternative proofs of two well known properties of free lattices: Whitman's condition and semidistributivity. It demonstrates usefulness of such proof systems outside logic.

A sequent system is used to give alternative proofs of two well known properties of free lattices: Whitman’s condition and semidistributivity. It demonstrates usefulness of such proof systems outside logic.

It is well known that the subvariety lattice of the variety of relation algebras has exactly three atoms. The (join-irreducible) covers of two of these atoms are known, but a complete classification of the (join-irreducible) covers of the remaining atom has not yet been found. These statements are also true of a related subvariety lattice, namely t...

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 f...

It is known that in the lattice of normal extensions of the logic KTB there are unique logics of codimensions 1 and 2, namely, the logic of a single reflexive point, and the logic of the total relation on two points. A natural question arises about the cardinality of the set of normal extensions of KTB of codimension 3. Generalising two finite exam...

We investigate a semigroup construction related to the two-sided wreath product. It encompasses a range of known constructions and gives a slightly finer version of the decomposition in the Krohn-Rhodes Theorem, in which the three-element flip-flop is replaced by the two-element semilattice. We develop foundations of the theory of our construction,...

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 algebr...

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 l...

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context, we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we observe a collapse in the applicability of algorithms for CSPs over directed graphs...

We show the admissibility for BCI of a rule form of the characteristic implicational axiom of abelian logic, this rule taking us from (α → β) → β to α. This is done in Section 8, with surrounding sections exploring the admissibility and derivability of various related rules in several extensions of BCI.

A qualitative representation $\phi$ is like an ordinary representation of a relation algebra, but instead of requiring $(a; b)^\phi = a^\phi | b^\phi$, as we do for ordinary representations, we only require that $c^\phi\supseteq a^\phi | b^\phi \iff c\geq a ; b$, for each $c$ in the algebra. A constraint network is qualitatively satisfiable if its...

We obtain representations for relation algebras corresponding to certain edge colourings of complete graphs. Suitable colourings are obtained for the number of colours n up to 120, with two exceptions: n = 8 and n = 13. For n > 7, it was not known whether representations exist.

Quasi-subtractive varieties are a generalization of subtractive varieties, introduced to account for some correspondence theorems between ideals and congruences in the literature that are not corollaries of general theorems in the theory of subtractive varieties. The main tool for the investigation of quasi-subtractive varieties is the concept of o...

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-discrimina...

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...

We exhibit a simple inference rule, which is admissible but not derivable in BCK, proving that BCK is not structurally complete. The argument is proof-theoretical.

We investigate the relation of independence between varieties, as well as a generalisation of such which we call strict quasi-independence. Concerning the former notion, we specify a procedure for constructing an independent companion of a given solvable subvariety of a congruence modular variety; we show that joins of independent varieties inherit...

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we prove the algebraic CSP dichotomy conjecture holds for digraphs whose symmetric closu...

Using Vaggione’s concept of central element in a double-pointed algebra, we introduce the notion of Boolean-like variety as a generalisation of Boolean algebras to an arbitrary similarity type. Appropriately relaxing the requirement that every element be central in any member of the variety, we obtain the more general class of semi-Boolean-like var...

We investigate a construction of a pseudo BL-algebra out of an $\ell$-group called a kite. We show that many well-known examples of algebras related to fuzzy logics can be obtained in that way. We describe subdirectly irreducible kites. As another application, we exhibit a new countably infinite family of varieties of pseudo BL-algebras covering th...

One of the central topics in computable algebra and model theory is concerned with the study of computable isomorphisms. Classically, we do not distinguish between isomorphic structures, however, from computability point of view, isomorphic structures can differ quite dramatically. A typical example is provided by the linear order of type ω. It has...

Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras. Abstract algebraic logic can explain these phenomena at a rat...

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.

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 a n = a n+1 holds...

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.

The main aim of the paper is to solve a problem posed in Di Nola et al. (Multiple Val. Logic 8:715–750, 2002) whether every pseudo BL-algebra with two negations is good, i.e. whether the two negations commute. This property is intimately connected with possessing a state, which in turn is essential in quantum logical applications. We approach the s...

We investigate GBL-algebras with the property that for every element a there is a positive integer n such that a^n = a^{n+1}. We show that varieties of GBL-algebras generated by such algebras are commutative. We also present examples showing that other conditions forcing commutativity will be hard to come by. 2000 Mathematics Subject Classificatio...

In the present paper we continue the investigation of the lattice of subvarieties of the variety of ${\sqrt{\prime}}$ quasi-MV algebras, already started in [6]. Beside some general results on the structure of such a lattice, the main contribution of this work is the solution of a long-standing open problem concerning these algebras: namely, we s...

Hájek's basic logic BL is an extension of the substructural logic FLew, or equivalently, Höhle's monoidal logic. Thus, fuzzy logics can be viewed as a special subclass of substructural logics. On the other hand, their close connections are often overlooked, since these two classes of logics have been motivated by different aims, and so introduced a...

Quasi-MV algebras are generalisations of MV algebras arising in quantum computational logic. Although a reasonably complete description of the lattice of subvarieties of quasi-MV algebras has already been provided, the problem of extending this description to the setting of quasivarieties has so far remained open. Given its apparent logical repercu...

In the present paper we continue the investigation of the lattice of subvarieties of the variety of Square-root-quasi-MV algebras, already started in [6]. Beside some general results on the structure of such a lattice, the main contribution of this work is the solution of a long-standing open problem concerning these algebras: namely, we show that...

We show that the variety of diassociative loops is not finitely based even relative to power associative loops with inverse property.

In the present paper, which is a sequel to [14] and [3], we investigate further the structure theory of quasi-MV algebras and root'quasi-MV algebras. In particular: we provide an improved version of the subdirect representation theorem for both varieties; we characterise the Ursini ideals of quasi-MV algebras; we establish a restricted version of J...

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....

We generalise in three different directions two well-known results in universal algebra. Grätzer, Lakser and Płonka proved that independent subvarieties V1, V2{\mathcal{V}_{1}, \mathcal{V}_{2}} of a variety V{\mathcal{V}} are disjoint and such that their join V1 ÚV2{\mathcal{V}_{1} \vee \mathcal{V}_{2}} (in the lattice of subvarieties of V{\mathcal...

Humberstone asks whether every theorem of BCI provably implies $\phi\to\phi$ for some formula $\phi$ . Meyer conjectures that the axiom $\mathbf{B}$ does not imply any such "self-implication." We prove a slightly stronger result, thereby confirming Meyer's conjecture.

We provide a simple sufficient criterion to show that a given variety of GBL-algebras does not admit (local) completions. As corollaries, we obtain that no variety of GBL-algebras containing Chang’s chain, no nontrivial variety of ℓ-groups, nor the variety of product algebras admit completions. The first result strengthens a result of Gehrke and Pr...

Constraint networks in qualitative spatial and temporal reasoning are always complete graphs. When one adds an extra el- ement to a given network, previously unknown constraints are de- rived by intersections and compositions of other constraints, and this may introduce inconsistency to the overall network. Likewise, when combining two consistent n...

It is proved that there are only two logics that split the lattice Next(KTB). The proof is based on the general splitting theorem by Kracht and conducted by a graph theoretic argument.

A b s t r a c t. It is shown that the pure (strict) implication fragment of the modal logic (3) has finitely many non-equivalent formulae in one variable. The exact number of such formulae is not known. We show that this finiteness result is the best possible, since the analogous fragment of S4, and therefore of (3), in two variables has infinitely...

The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is...

Maddux observed a tantalisingly close connection between certain relation algebras and relevant logics R and RM. He asks whether this connection amounts to full interpretability. Although unable to answer that question, we prove that a version of positive minimal relevant logic B is fully interpretable in the variety of weakly associative relation...

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.

In this paper we show that a variety of modal algebras of finite type is semisimple iff it is discriminator iff it is both weakly transitive and cyclic. This fact has been claimed already in [4] (based on joint work by the two authors) but the proof was fatally flawed.

We present constructions producing continua of almost minimal subvarieties of certain varieties related to fuzzy logic. We also prove that there are only countably many almost minimal varieties of Hajek’s BL-algebras – all of them rather well-known. Some contrasting results on varieties satisfying the 2-potency condition x 3 =x 2 are also included....

In this journal I published a paper [ibid. 67, No. 3, 933–946 (2002; Zbl 1013.03029)] entitled “PDL has interpolation” purporting to prove what the title announced. It has been pointed out to me by Yde Venema that my argument contains a serious error. As I have not been able to correct it, the problem of interpolation for Propositional Dynamic Logi...

We prove that all semisimple varieties of FLew-algebras are discriminator va- rieties. A characterisation of discriminator and EDPC varieties of FLew-algebras follows. It matches exactly a natural classication of logics over FLew proposed by H. Ono.

McNaughton in his known paper [7], motivated by the work of Gurevich and Harrington [4], introduced a class of games played on finite graphs. In his paper McNaughton proves that winners in his games have winning strategies that can be implemented by finite state automata. McNaughton games have attracted attention of many experts in the area, partly...

It is proved that free dynamic algebras superamalgamate. Craig interpolation for proposi- tional dynamic logic and superamalgamation for the variety of dynamic algebras follow.

We construct a variety of tense algebras that is not generated by its atomic members. Then we lift this result to the case of modal algebras.

The interpolation theorem for propositional dynamic logic is proved, solving a long-standing open question in the area.

. It is shown that the only algebra that splits the lattice of subvarieties of the variety of residuated lattices is the two element boolean algebra.

A simple algebraic framework is constructed, in which nonstochastic GHZ-Belltheorems can be analyzed. The framework merges Belnap''s outcomes in branchingtime with his branching space-time (BST). We show that an important structurein BST, called the family of outcomes of an event, is a Boolean algebra. We provethat there is no common cause that acc...

A class K of algebras will be said to have the congruence extension property (CEP) iff for any algebras A, B from K and any congruence Θ ∈ ConA, we have: if A ⊆ B, then there is a Φ ∈ ConB with Θ = Φ| A. This property is sometimes called the absolute CEP. A congruence Θ on a member A of K is called a K-congruence iff A/Θ belongs to K as well. Con K...

We show that the variety of residuated lattices is generated by its finite simple members, improving upon a finite model property result of Okada and Terui. The reasoning is a blend of proof-theoretic and algebraic arguments.

The paper intends to provide an algebraic framework in which subluminal causation can be analysed. The framework merges Belnap’s ‘outcomes in branching time’ with his ‘branching space-time’ (BST). It is shown that an important structure in BST, called ‘family of outcomes of an event’, is a boolean algebra. We define next non-stochastic common cause...

A b s t r a c t. The paper has two parts preceded by quite compre- hensive preliminaries. In the rst part it is shown that a subvariety of the variety T of all tense algebras is discriminator if and only if it is semisimple. The variety T turns out to be the join of an increasing chain of varieties Dn, which are discriminator varieties. The argumen...

A syntactic derivation of the Cornish identity (J) from the axioms of HBCK is presented which amounts to a syntactic proof of Wroński’s conjecture that the naturally ordered BCK-algebras form a variety.

A b s t r a c t. We present constructions producing continua of almost minimal subvarieties of certain varieties related to fuzzy logic. We also prove that there are only countably many almost minimal varieties of Hajek's BL-algebras | all of them rather well known. Some contrasting results on varieties satisfying the 2-potency condition x3 = x2 ar...

We show that Bruns and Harding's counterexample (see [1]) to amalgamation in orthomodular lattices also works for Hilbert lattices. The argument is based on the example of a non-modular Hilbert lattice devised by von Neumann in a letter to Birkhoff (see [2] for an extensive quotation from that letter).