999 research outputs found
Congruences and ideals in lattice effect algebras as basic algebras
summary:Effect basic algebras (which correspond to lattice ordered effect algebras) are studied. Their ideals are characterized (in the language of basic algebras) and one-to-one correspondence between ideals and congruences is shown. Conditions under which the quotients are OMLs or MV-algebras are 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
Lattice congruences, fans and Hopf algebras
We give a unified explanation of the geometric and algebraic properties of
two well-known maps, one from permutations to triangulations, and another from
permutations to subsets. Furthermore we give a broad generalization of the
maps. Specifically, for any lattice congruence of the weak order on a Coxeter
group we construct a complete fan of convex cones with strong properties
relative to the corresponding lattice quotient of the weak order. We show that
if a family of lattice congruences on the symmetric groups satisfies certain
compatibility conditions then the family defines a sub Hopf algebra of the
Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has
a basis which is described by a type of pattern-avoidance. Applying these
results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite
sequence of smaller algebras, where the second algebra in the sequence is the
Hopf algebra of non-commutative symmetric functions. We also associate both a
fan and a Hopf algebra to a set of permutations which appears to be
equinumerous with the Baxter permutations.Comment: 34 pages, 1 figur
Lattices of quasi-equational theories as congruence lattices of semilattices with operators, Part I
We show that for every quasivariety K of structures (where both functions and
relations are allowed) there is a semilattice S with operators such that the
lattice of quasi-equational theories of K (the dual of the lattice of
sub-quasivarieties of K) is isomorphic to Con(S,+,0,F). As a consequence, new
restrictions on the natural quasi-interior operator on lattices of
quasi-equational theories are found.Comment: Presented on International conference "Order, Algebra and Logics",
Vanderbilt University, 12-16 June, 2007 25 pages, 2 figure
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