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Implication and equivalential reducts of basic algebras

Ivan Chajda, Miroslav Kolařík, Filip Švrček (2010)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

A term operation implication is introduced in a given basic algebra 𝒜 and properties of the implication reduct of 𝒜 are treated. We characterize such implication basic algebras and get congruence properties of the variety of these algebras. A term operation equivalence is introduced later and properties of this operation are described. It is shown how this operation is related with the induced partial order of 𝒜 and, if this partial order is linear, the algebra 𝒜 can be reconstructed by means of...

Join-semilattices with two-dimensional congruence amalgamation

Friedrich Wehrung (2002)

Colloquium Mathematicae

We say that a ⟨∨,0⟩-semilattice S is conditionally co-Brouwerian if (1) for all nonempty subsets X and Y of S such that X ≤ Y (i.e. x ≤ y for all ⟨x,y⟩ ∈ X × Y), there exists z ∈ S such that X ≤ z ≤ Y, and (2) for every subset Z of S and all a, b ∈ S, if a ≤ b ∨ z for all z ∈ Z, then there exists c ∈ S such that a ≤ b ∨ c and c ≤ Z. By restricting this definition to subsets X, Y, and Z of less than κ elements, for an infinite cardinal κ, we obtain the definition of a conditionally κ-co-Brouwerian...

J-subspace lattices and subspace M-bases

W. Longstaff, Oreste Panaia (2000)

Studia Mathematica

The class of J-lattices was defined in the second author’s thesis. A subspace lattice on a Banach space X which is also a J-lattice is called a J- subspace lattice, abbreviated JSL. Every atomic Boolean subspace lattice, abbreviated ABSL, is a JSL. Any commutative JSL on Hilbert space, as well as any JSL on finite-dimensional space, is an ABSL. For any JSL ℒ both LatAlg ℒ and ℒ ⊥ (on reflexive space) are JSL’s. Those families of subspaces which arise as the set of atoms of some JSL on X are characterised...

Lattice effect algebras densely embeddable into complete ones

Zdena Riečanová (2011)

Kybernetika

An effect algebraic partial binary operation ø p l u s defined on the underlying set E uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion E ^ of E there exists an effect algebraic partial binary operation ⊕ ^ then ⊕ ^ need not be an extension of ⊕ . Moreover, for an Archimedean atomic lattice effect algebra E we give a necessary and sufficient condition for that ⊕ ^ existing on E ^ is an extension of ⊕ defined on E . Further we show that such ⊕ ^ extending ⊕ exists at most...

Lattices with complemented tolerance lattice

Sándor Radelecki, Dietmar Schweigert (2004)

Czechoslovak Mathematical Journal

We characterize lattices with a complemented tolerance lattice. As an application of our results we give a characterization of bounded weakly atomic modular lattices with a Boolean tolerance lattice.

Lyapunov measures on effect algebras

Anna Avallone, Giuseppina Barbieri (2003)

Commentationes Mathematicae Universitatis Carolinae

We prove a Lyapunov type theorem for modular measures on lattice ordered effect algebras.

Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras

Martin Kalina (2010)

Kybernetika

If element z of a lattice effect algebra ( E , ⊕ , 0 , 1 ) is central, then the interval [ 0 , z ] is a lattice effect algebra with the new top element z and with inherited partial binary operation ⊕ . It is a known fact that if the set C ( E ) of central elements of E is an atomic Boolean algebra and the supremum of all atoms of C ( E ) in E equals to the top element of E , then E is isomorphic to a subdirect product of irreducible effect algebras ([18]). This means that if there exists a MacNeille completion E ^ of E which is its extension...

Minimal bounded lattices with an antitone involution the complemented elements of which do not form a sublattice

Ivan Chajda, Helmut Länger (2008)

Discussiones Mathematicae - General Algebra and Applications

Bounded lattices with an antitone involution the complemented elements of which do not form a sublattice must contain two complemented elements such that not both their join and their meet are complemented. We distinguish (up to symmetry) eight cases and in each of these cases we present such a lattice of minimal cardinality.

Currently displaying 81 – 100 of 217