248 research outputs found
Kite Pseudo Effect Algebras
We define a new class of pseudo effect algebras, called kite pseudo effect
algebras, which is connected with partially ordered groups not necessarily with
strong unit. In such a case, starting even with an Abelian po-group, we can
obtain a noncommutative pseudo effect algebra. We show how such kite pseudo
effect algebras are tied with different types of the Riesz Decomposition
Properties. Kites are so-called perfect pseudo effect algebras, and we define
conditions when kite pseudo effect algebras have the least non-trivial normal
ideal
On a New Construction of Pseudo Effect Algebras
We define a new class of pseudo effect algebras, called kite pseudo effect
algebras, which is connected not necessarily with partially ordered groups, but
rather with generalized pseudo effect algebras where the greatest element is
not guaranteed. Starting even with a commutative generalized pseudo effect
algebra, we can obtain a non-commutative pseudo effect algebra. We show how
such kite pseudo effect algebras are tied with different types of the Riesz
Decomposition Properties. We find conditions when kite pseudo effect algebras
have the least non-trivial normal ideal.Comment: arXiv admin note: substantial text overlap with arXiv:1306.030
Pseudo MV-algebras and Lexicographic Product
We study algebraic conditions when a pseudo MV-algebra is an interval in the
lexicographic product of an Abelian unital -group and an -group
that is not necessary Abelian. We introduce -perfect pseudo MV-algebras
and strong -perfect pseudo MV-algebras, the latter ones will have a
representation by a lexicographic product. Fixing a unital -group
, the category of strong -perfect pseudo MV-algebras is
categorically equivalent to the category of -groups.Comment: arXiv admin note: text overlap with arXiv:1304.074
Lattice of closure endomorphisms of a Hilbert algebra
A closure endomorphism of a Hilbert algebra A is a mapping that is
simultaneously an endomorphism of and a closure operator on A. It is known that
the set CE of all closure endomorphisms of A is a distributive lattice where
the meet of two elements is defined pointwise and their join is given by their
composition. This lattice is shown in the paper to be isomorphic to the lattice
of certain filters of A, anti-isomorphic to the lattice of certain closure
retracts of A, and compactly generated. The set of compact elements of CE
coincides with the adjoint semilattice of A, conditions under which two Hilbert
algebras have isomorphic adjoint semilattices (equivalently, minimal Brouwerian
extensions) are discussed. Several consequences are drawn also for implication
algebras.Comment: 16 pages, no figures, submitted to Algebra Universalis (under review
since 24.11.2015
Noncommutative lattices
The extended study of non-commutative lattices was begun in 1949 by Ernst Pascual Jordan, a theoretical and mathematical physicist and co-worker of Max Born and Werner Karl Heisenberg. Jordan introduced noncommutative lattices as algebraic structures potentially suitable to encompass the logic of the quantum world. The modern theory of noncommutative lattices began 40 years later with Jonathan Leech\u27s 1989 paper "Skew lattices in rings." Recently, noncommutative generalizations of lattices and related structures have seen an upsurge in interest, with new ideas and applications emerging, from quasilattices to skew Heyting algebras. Much of this activity is derived in some way from the initiation, over thirty years ago, of Jonathan Leech\u27s program of research that studied noncommutative variations of lattices. The present book consists of seven chapters, mainly covering skew lattices, quasilattices and paralattices, skew lattices of idempotents in rings and skew Boolean algebras. As such, it is the first research monograph covering major results due to the renewed study of noncommutative lattices. It will serve as a valuable graduate textbook on the subject, as well as handy reference to researchers of noncommutative algebras
Quantum-BE algebras
We redefine the quantum-MV algebras starting from involutive BE algebras and
we introduce and study the notion of quantum-BE algebras. We define the
commutative quantum-BE algebras and we give a characterization of these
structures. We also prove that any bounded commutative BCK algebra is a
quantum-BE algebra, and give conditions for quantum-BE algebras to be bounded
commutative BCK algebras. Furthermore, we prove that bounded commutative BCK
algebras are both quantum-BE algebras and quantum-B algebras. Finally, we
provide certain conditions for quantum-MV algebras to be MV algebras
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