123 research outputs found
Independent axiom systems for nearlattices
A nearlattice is a join semilattice such that every principal filter is a
lattice with respect to the induced order. Hickman and later Chajda et al
independently showed that nearlattices can be treated as varieties of algebras
with a ternary operation satisfying certain axioms. Our main result is that the
variety of nearlattices is 2-based, and we exhibit an explicit axiom system of
two independent identities. We also show that the original axiom systems of
Hickman and of Chajda et al are, respectively, dependent.Comment: 16 pages in 12pt; v2: minor changes suggested by referee, to appear
in Czechoslovak Math.
The groupoid-based logic for lattice effect algebras
The aim of the paper is to establish a certain logic corresponding to lattice
effect algebras. First, we answer a natural question whether a lattice effect
algebra can be represented by means of a groupoid-like structure. We establish
a one-to-one correspondence between lattice effect algebras and certain
groupoids with an antitone involution. Using these groupoids, we are able to
introduce a suitable logic for lattice effect algebras.Comment: 7 page
Varieties of distributive rotational lattices
A rotational lattice is a structure (L;\vee,\wedge, g) where
L=(L;\vee,\wedge) is a lattice and g is a lattice automorphism of finite order.
We describe the subdirectly irreducible distributive rotational lattices. Using
J\'onsson's lemma, this leads to a description of all varieties of distributive
rotational lattices.Comment: 7 page
Representing quantum structures as near semirings
In this article, we introduce the notion of near semiring with involution. Generalizing the theory of semirings we aim at represent quantum structures, such as basic algebras and orthomodular lattices, in terms of near semirings with involution. In particular, after discussing several properties of near semirings, we introduce the so-called Łukasiewicz near semirings, as a particular case of near semirings, and we show that every basic algebra is representable as (precisely, it is term equivalent to) a near semiring. In the particular case in which a Łukasiewicz near semiring is also a semiring, we obtain as a corollary a representation of MV-algebras as semirings. Analogously, by introducing a particular subclass of Łukasiewicz near semirings, that we termed orthomodular near semirings, we obtain a representation of orthomodular lattices. In the second part of the article, we discuss several universal algebraic properties of Łukasiewicz near semirings and we show that the variety of involutive integral near semirings is a Church variety. This yields a neat equational characterization of central elements of this variety. As a byproduct of such, we obtain several direct decomposition theorems for this class of algebras
Normalization of -algebras
summary:We consider algebras determined by all normal identities of -algebras, i.e. algebras of many-valued logics. For such algebras, we present a representation based on a normalization of a sectionally involutioned lattice, i.e. a -lattice, and another one based on a normalization of a lattice-ordered group
Lattice congruences of the weak order
We study the congruence lattice of the poset of regions of a hyperplane
arrangement, with particular emphasis on the weak order on a finite Coxeter
group. Our starting point is a theorem from a previous paper which gives a
geometric description of the poset of join-irreducibles of the congruence
lattice of the poset of regions in terms of certain polyhedral decompositions
of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let
\eta_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show
that the fibers of \eta_K constitute the smallest lattice congruence with
1\equiv s for every s\in(S-K). We give an algorithm for determining the
congruence lattice of the weak order for any finite Coxeter group and for a
finite Coxeter group of type A or B we define a directed graph on subsets or
signed subsets such that the transitive closure of the directed graph is the
poset of join-irreducibles of the congruence lattice of the weak order.Comment: 26 pages, 4 figure
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