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Descriptions of state spaces of orthomodular lattices (the hypergraph approach)

Mirko Navara (1992)

Mathematica Bohemica

Using the general hypergraph technique developed in [7], we first give a much simpler proof of Shultz's theorem [10]: Each compact convex set is affinely homeomorphic to the state space of an orthomodular lattice. We also present partial solutions to open questions formulated in [10] - we show that not every compact convex set has to be a state space of a unital orthomodular lattice and that for unital orthomodular lattices the state space characterization can be obtained in the context of unital...

Didactical note: probabilistic conditionality in a Boolean algebra.

Enric Trillas, Claudi Alsina, Settimo Termini (1996)

Mathware and Soft Computing

This note deals with two logical topics and concerns Boolean Algebras from an elementary point of view. First we consider the class of operations on a Boolean Algebra that can be used for modelling If-then propositions. These operations, or Conditionals, are characterized under the hypothesis that they only obey to the Modus Ponens-Inequality, and it is shown that only six of them are boolean two-place functions. Is the Conditional Probability the Probability of a Conditional? This problem will...

Dimension in algebraic frames, II: Applications to frames of ideals in

Jorge Martinez, Eric R. Zenk (2005)

Commentationes Mathematicae Universitatis Carolinae

This paper continues the investigation into Krull-style dimensions in algebraic frames. Let be an algebraic frame. is the supremum of the lengths of sequences of (proper) prime elements of . Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of in terms of the dimensions of certain boundary quotients of . This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily...

Direct product decompositions of bounded commutative residuated -monoids

Ján Jakubík (2008)

Czechoslovak Mathematical Journal

The notion of bounded commutative residuated -monoid (-monoid, in short) generalizes both the notions of -algebra and of -algebra. Let be a -monoid; we denote by the underlying lattice of . In the present paper we show that each direct...

Directoids with sectionally antitone involutions and skew MV-algebras

Ivan Chajda, Miroslav Kolařík (2007)

Mathematica Bohemica

It is well-known that every MV-algebra is a distributive lattice with respect to the induced order. Replacing this lattice by the so-called directoid (introduced by J. Ježek and R. Quackenbush) we obtain a weaker structure, the so-called skew MV-algebra. The paper is devoted to the axiomatization of skew MV-algebras, their properties and a description of the induced implication algebras.

Directoids with sectionally switching involutions

Ivan Chajda (2006)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

It is shown that every directoid equipped with sectionally switching mappings can be represented as a certain implication algebra. Moreover, if the directoid is also commutative, the corresponding implication algebra is defined by four simple identities.

Discriminator varieties of Boolean algebras with residuated operators

Peter Jipsen (1993)

Banach Center Publications

The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral r-algebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated...

Distributive implication groupoids

Ivan Chajda, Radomir Halaš (2007)

Open Mathematics

We introduce a concept of implication groupoid which is an essential generalization of the implication reduct of intuitionistic logic, i.e. a Hilbert algebra. We prove several connections among ideals, deductive systems and congruence kernels which even coincide whenever our implication groupoid is distributive.

DMF-algebras: representation and topological characterization

Maurizio Negri (1998)

Bollettino dell'Unione Matematica Italiana

Gli insiemi parziali sono coppie di sottoinsiemi di , dove . Gli insiemi parziali su costituiscono una DMF-algebra, ossia un'algebra di De Morgan in cui la negazione ha un solo punto fisso. Dimostriamo che ogni DMF-algebra è isomorfa a un campo di insiemi parziali. Utilizzando gli insiemi parziali su come aperti, introduciamo il concetto di spazio topologico parziale su . Infine associamo ad ogni DMF-algebra uno spazio topologico parziale i cui clopen compatti costituiscono un campo d'insiemi...

Duality for Hilbert algebras with supremum: An application

Hernando Gaitan (2017)

Mathematica Bohemica

We modify slightly the definition of -partial functions given by Celani and Montangie (2012); these partial functions are the morphisms in the category of -space and this category is the dual category of the category with objects the Hilbert algebras with supremum and morphisms, the algebraic homomorphisms. As an application we show that finite pure Hilbert algebras with supremum are determined by the monoid of their endomorphisms.

Dually residuated -monoids having no non-trivial convex subalgebras

Jan Kühr (2006)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In this note we describe the structure of dually residuated -monoids (-monoids) that have no non-trivial convex subalgebras.

Effect algebras and ring-like structures

Enrico G. Beltrametti, Maciej J. Maczyński (2003)

Discussiones Mathematicae - General Algebra and Applications

The dichotomic physical quantities, also called propositions, can be naturally associated to maps of the set of states into the real interval [0,1]. We show that the structure of effect algebra associated to such maps can be represented by quasiring structures, which are a generalization of Boolean rings, in such a way that the ring operation of addition can be non-associative and the ring multiplication non-distributive with respect to addition. By some natural assumption on the effect algebra,...

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