Expanding Quasi-MV Algebras by a Quantum Operator

We investigate an expansion of quasi-MV algebras ([10]) by a genuine quantum unary operator. The variety of such quasi-MV algebras has a subquasivariety whose members—called cartesian—can be obtained in an appropriate way out of MV algebras. After showing that cartesian . quasi-MV algebras generate ,we prove a standard completeness theorem for w.r.t. an algebra over the complex numbers. Presented by Heinrich Wansing     

Decomposability of the Finitely Generated Free Hoop Residuation Algebra

In this paper we prove that, for n > 1, the n-generated free algebra in any locally finite subvariety of HoRA can be written in a unique nontrivial way as Ł₂ × A′, where A′ is a directly indecomposable algebra in . More precisely, we prove that the unique nontrivial pair of factor congruences of is given by the filters and , where the element is recursively defined from the term introduced by W. H. Cornish. As an additional result we obtain a characterization of minimal irreducible filters of in terms of its coatoms. Presented by Daniele Mundici     

A Note on Neat Reducts

SC, CA, QA and QEA denote the class of Pinter’s substitution algebras, Tarski’s cylindric algebras, Halmos’ quasi-polyadic and quasi-polyadic equality algebras, respectively. Let . and . We show that the class of n dimensional neat reducts of algebras in K _m is not elementary. This solves a problem in [2]. Also our result generalizes results proved in [1] and [2]. Presented by Robert Goldblatt     

Congruence Coherent Symmetric Extended de Morgan Algebras

An algebra A is said to be congruence coherent if every subalgebra of A that contains a class of some congruence on A is a union of -classes. This property has been investigated in several varieties of lattice-based algebras. These include, for example, de Morgan algebras, p-algebras, double p-algebras, and double MS-algebras. Here we determine precisely when the property holds in the class of symmetric extended de Morgan algebras. Presented by M.E. Adams     

Equivalents for a Quasivariety to be Generated by a Single Structure

We present some equivalent conditions for a quasivariety of structures to be generated by a single structure. The first such condition, called the embedding property was found by A.I. Mal′tsev in [6]. It says that if are nontrivial, then there exists such that A and B are embeddable into C. One of our equivalent conditions states that the set of quasi-identities valid in is closed under a certain Gentzen type rule which is due to J. Łoś and R. Suszko [5]. Presented by Jacek Malinowski     

A Propositional Logic with Relative Identity Connective and a Partial Solution to the Paradox of Analysis

We construct a a system PLRI which is the classical propositional logic supplied with a ternary construction , interpreted as the intensional identity of statements and in the context . PLRI is a refinement of Roman Suszko’s sentential calculus with identity (SCI) whose identity connective is a binary one. We provide a Hilbert-style axiomatization of this logic and prove its soundness and completeness with respect to some algebraic models. We also show that PLRI can be used to give a partial solution to the paradox of analysis. Presented by Jacek Malinowski     

The Variety of Lattice Effect Algebras Generated by MV-algebras and the Horizontal Sum of Two 3-element Chains

It has been recently shown [4] that the lattice effect algebras can be treated as a subvariety of the variety of so-called basic algebras. The open problem whether all subdirectly irreducible distributive lattice effect algebras are just subdirectly irreducible MV-chains and the horizontal sum of two 3-element chains is in the paper transferred into a more tractable one. We prove that modulo distributive lattice effect algebras, the variety generated by MV-algebras and is definable by three simple identities and the problem now is to check if these identities are satisfied by all distributive lattice effect algebras or not. Presented by Daniele Mundici     

Finite basis problems and results for quasivarieties

Let be a finite collection of finite algebras of finite signature such that SP( ) has meet semi-distributive congruence lattices. We prove that there exists a finite collection ₁ of finite algebras of the same signature, , such that SP( ₁) is finitely axiomatizable.We show also that if , then SP( ₁) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract indicates.While working on this paper, the first author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant no. T37877 and the second author was supported by the US National Science Foundation grant no. DMS-0245622.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

An n-Player Semantic Game for an n + 1-Valued Logic

First we show that the classical two-player semantic game actually corresponds to a three-valued logic. Then we generalize this result and give an n-player semantic game for an n + 1-valued logic with n binary connectives, each associated with a player. We prove that player i has a winning strategy in game if and only if the truth value of is t _i in the model M, for 1 ≤ i ≤ n; and none of the players has a winning strategy in if and only if the truth value of is t ₀ in M.     

An algebraic study of well-foundedness

A foundational algebra ( , f, ) consists of a hemimorphism f on a Boolean algebra with a greatest solution to the condition f(x). The quasi-variety of foundational algebras has a decidable equational theory, and generates the same variety as the complex algebras of structures (X, R), where f is given by R-images and is the non-wellfounded part of binary relation R.The corresponding results hold for algebras satisfying =0, with respect to complex algebras of wellfounded binary relations. These algebras, however, generate the variety of all ( ,f) with f a hemimorphism on ).Admitting a second hemimorphism corresponding to the transitive closure of R allows foundational algebras to be equationally defined, in a way that gives a refined analysis of the notion of diagonalisable algebra.The research reported in this paper was carried out while the author was at the Department of Mathematics and Statistics, University of Auckland, under a teaching exchange with the Victoria University of Wellington.     

Finitistic and Frequentistic Approximation of Probability Measures with or without <Emphasis Type="Italic">σ</Emphasis>-Additivity

In this paper a theory of finitistic and frequentistic approximations — in short: f-approximations — of probability measures P over a countably infinite outcome space N is developed. The family of subsets of N for which f-approximations converge to a frequency limit forms a pre-Dynkin system . The limiting probability measure over D can always be extended to a probability measure over , but this measure is not always σ-additive. We conclude that probability measures can be regarded as idealizations of limiting frequencies if and only if σ-additivity is not assumed as a necessary axiom for probabilities. We prove that σ-additive probability measures can be characterized in terms of so-called canonical and in terms of so-called full f-approximations. We also show that every non-σ-additive probability measure is f-approximable, though neither canonically nor fully f-approximable. Finally, we transfer our results to probability measures on open or closed formulas of first-order languages.     

Dominions in quasivarieties of universal algebras

The dominion of a subalgebra H in an universal algebra A (in a class ) is the set of all elements such that for all homomorphisms if f, g coincide on H, then a^f = a^g. We investigate the connection between dominions and quasivarieties. We show that if a class is closed under ultraproducts, then the dominion in is equal to the dominion in a quasivariety generated by . Also we find conditions when dominions in a universal algebra form a lattice and study this lattice.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

Proper Semantics for Substructural Logics,from a Stalker Theoretic Point of View

We study filters in residuated structures that are associated with congruence relations (which we call -filters), and develop a semantical theory for general substructural logics based on the notion of primeness for those filters. We first generalize Stone’s sheaf representation theorem to general substructural logics and then define the primeness of -filters as being “points” (or stalkers) of the space, the spectrum, on which the representing sheaf is defined. Prime FL-filters will turn out to coincide with truth sets under various well known semantics for certain substructural logics. We also investigate which structural rules are needed to interpret each connective in terms of prime -filters in the same way as in Kripke or Routley-Meyer semantics. We may consider that the set of the structural rules that each connective needs in this sense reflects the difficulty of giving the meaning of the connective. A surprising discovery is that connectives , ⅋ of linear logic are linearly ordered in terms of the difficulty in this sense. Presented by Wojciech Buszkowski     

A theorem in 3-valued model theory with connections to number theory,type theory,and relevant logic

Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for number theory was obtained earlier by R. K. Meyer and suggested the present abstract development).     

Modal Incompleteness Revisited

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some L and some cannot be separated from by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any class of complete BAOs. Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Löb logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind.     

Frames and MV-algebras

We describe a class of MV-algebras which is a natural generalization of the class of “algebras of continuous functions”. More specifically, we're interested in the algebra of frame maps Hom (Ω(A), K) in the category T of frames, where A is a topological MV-algebra, Ω(A) the lattice of open sets of A, and K an arbitrary frame. Given a topological space X and a topological MV-algebra A, we have the algebra C (X, A) of continuous functions from X to A. We can look at this from a frame point of view. Among others we have the result: if K is spatial, then C(pt(K), A), pt(K) the points of K, embeds into Hom (Ω(A), K) analogous to the case of C (X, A) embedding into Hom (Ω(A), Ω (X)). 1991 Mathematics Subject Classification: 06F20, 06F25, 06D30 Presented by Ewa Orlowska     

Moral Conflicts between Groups of Agents

Two groups of agents, and , face a moral conflict if has a moral obligation and has a moral obligation, such that these obligations cannot both be fulfilled. We study moral conflicts using a multi-agent deontic logic devised to represent reasoning about sentences like ‘In the interest of group of agents, group of agents ought to see to it that .’ We provide a formal language and a consequentialist semantics. An illustration of our semantics with an analysis of the Prisoner’s Dilemma follows. Next, necessary and sufficient conditions are given for (1) the possibility that a single group of agents faces a moral conflict, for (2) the possibility that two groups of agents face a moral conflict within a single moral code, and for (3) the possibility that two groups of agents face a moral conflict.     

On Neat Reducts of Algebras of Logic

SC , CA , QA and QEA stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension , respectively. Generalizing a result of Németi on cylindric algebras, we show that for K {SC, CA, QA, QEA} and ordinals < , the class Nr K of -dimensional neat reducts of -dimensional K algebras, though closed under taking homomorphic images and products, is not closed under forming subalgebras (i.e. is not a variety) if and only if > 1.From this it easily follows that for 1 < < , the operation of forming -neat reducts of algebras in K does not commute with forming subalgebras, a notion to be made precise.We give a contrasting result concerning Halmos' polyadic algebras (with and without equality). For such algebras, we show that the class of infinite dimensional neat reducts forms a variety.We comment on the status of the property of neat reducts commuting with forming subalgebras for various reducts of polyadic algebras that are also expansions of cylindric-like algebras. We try to draw a borderline between reducts that have this property and reducts that do not.Following research initiated by Pigozzi, we also emphasize the strong tie that links the (apparently non-related) property of neat reducts commuting with forming subalgebras with proving amalgamation results in cylindric-like algebras of relations. We show that, like amalgamation, neat reducts commuting with forming subalgebras is another algebraic expression of definability and, accordingly, is also strongly related to the well-known metalogical properties of Craig, Beth and Robinson in the corresponding logics.     

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