共查询到20条相似文献,搜索用时 31 毫秒
1.
The aim of this paper is to present several techniques of constructing a lattice-ordered effect algebra from a given family of lattice-ordered effect algebras, and to study the structure of finite lattice-ordered effect algebras. Firstly, we prove that any finite MV-effect algebra can be obtained by substituting the atoms of some Boolean algebra by linear MV-effect algebras. Then some conditions which can guarantee that the pasting of a family of effect algebras is an effect algebra are provided. At last, we prove that any finite lattice-ordered effect algebra E without atoms of type 2 can be obtained by substituting the atoms of some orthomodular lattice by linear MV-effect algebras. Furthermore, we give a way how to paste a lattice-ordered effect algebra from the family of MV-effect algebras. 相似文献
2.
Jingjing Ma 《代数通讯》2013,41(5):1680-1689
The main result in this article is to show that a regular unital finite-dimensional lattice-ordered algebra over ? with zero ?-radical is isomorphic to a finite direct sum of lattice-ordered matrix algebras of lattice-ordered group algebras of finite groups over ?. 相似文献
3.
In this paper, characterizations are given for the free lattice-ordered group over a generalized Boolean algebra and the freel -module of a totally ordered integral domain with unit over a generalized Boolean algebra. Extensions of lattice-ordered groups using generalized Boolean algebras are defined and their properties studied. 相似文献
4.
Let A be a lattice-ordered algebra endowed with a topology compatible with the structure of algebra. We provide internal conditions for A to be isomorphic as lattice-ordered algebras and homeomorphic to Ck(X), the lattice-ordered algebra C(X) of real continuous functions on a completely regular and Hausdorff topological space X, endowed with the topology of uniform convergence on compact sets. As a previous step, we determine this topology among the locally m-convex topologies on C(X) with the property that each order closed interval is bounded. 相似文献
5.
Jingjing Ma 《Algebra Universalis》2011,65(4):341-351
It is shown that a unital finite-dimensional ℓ-simple ℓ-algebra with a distributive basis is isomorphic to a lattice-ordered matrix algebra with the entrywise lattice order over
a lattice-ordered twisted group algebra of a finite group with the coordinatewise lattice order. It is also shown that the
isomorphism is unique. 相似文献
6.
We construct all the lattice orders (up to isomorphism) on a full matrix algebra over a subfield of the field of real numbers
so that it becomes a lattice-ordered algebra.
Received June 26, 2001; accepted in final form February 9, 2002. 相似文献
7.
For an n ×n matrix algebra over a totally ordered integral domain, necessary and sufficient conditions are derived such that the entrywise lattice order on it is the only lattice order (up to an isomorphism) to make it into a lattice-ordered algebra in which the identity matrix is positive. The conditions are then applied to particular integral domains. In the second part of the paper we consider n ×n matrix rings containing a positive n-cycle over totally ordered rings. Finally a characterization of lattice-ordered matrix ring with the entrywise lattice order is given. 相似文献
8.
Jingjing Ma 《Algebra Universalis》2009,62(1):51-61
We construct all the lattice orders on a 2 × 2 triangular matrix algebra over a totally ordered field that make it into a
lattice-ordered algebra. It is shown that every lattice order in which the identity matrix is not positive may be obtained
from a lattice order in which the identity matrix is positive. 相似文献
9.
We show that the quotient of a dimension effect algebra by its dimension equivalence relation is a unital bounded lattice-ordered
positive partial abelian monoid that satisfies a version of the Riesz decomposition property. For a dimension effect algebra
of finite type, the quotient is a centrally orthocomplete Stone–Heyting MV-effect algebra; moreover, an orthocomplete effect
algebra in which equality is a dimension equivalence relation is the same thing as a complete Stone–Heyting MV-effect algebra. 相似文献
10.
The lattice of monotonely Cauchy (=pre-Lebesgue) locally solid topologies on a given lattice-ordered group is studied. Indentifying topologies agreeing on order bounded sets this lattice becomes a complete Boolean algebra isomorphic to the subalgebra of the lattice's complemented members and realizable as a Boolean algebra of order projections. Some consequences of these results are indicated.Work done while Tim Traynor was visiting professor at University of Napoli sponsored by CNR-Italia. 相似文献
11.
Murray Neuzerling 《Algebra Universalis》2016,76(4):431-443
We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with respect to a universal relation, this result can be extended to infinite representations of ordered complemented semigroups. 相似文献
12.
《Quaestiones Mathematicae》2013,36(6):835-839
AbstractLet C (X) denote the lattice-ordered algebra of all real-valued continuous functions on a topological space X. This paper discusses in Zermelo-Fraenkel Set Theory the equivalence on C (X) between algebra homomorphisms, lattice homo- morphisms, and point evaluations. 相似文献
13.
Let A be a lattice-ordered group, B a generalized Boolean algebra. The Boolean extension A B of A has been investigated in the literature; we will refer to A B as a generalized Specker lattice-ordered group (namely, if A is the linearly ordered group of all integers, then A B is a Specker lattice-ordered group). The paper establishes that some distributivity laws extend from A B to both A and B, and (under certain circumstances) also conversely. 相似文献
14.
In [3] it was shown that a (real) signed measure on a cyclic coarse-grained quantum logic can be extended, as a signed measure,
over the entire power algebra. Later ([9]) this result was re-proved (and further improved on) and, moreover, the non-negative
measures were shown to allow for extensions as non-negative measures. In both cases the proof technique used was the technique
of linear algebra. In this paper we further generalize the results cited by extending group-valued measures on cyclic coarse-grained
quantum logics (or non-negative group-valued measures for lattice-ordered groups). Obviously, the proof technique is entirely
different from that of the preceding papers. In addition, we provide a new combinatorial argument for describing all atoms
of cyclic coarse-grained quantum logics. 相似文献
15.
16.
In the theory of lattice-ordered groups, there are interesting examples of properties — such as projectability — that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semi-linear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a,1] is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety. 相似文献
17.
Giovanni Panti 《代数通讯》2013,41(8):2849-2861
MV-algebras can be viewed either as the Lindenbaum algebras of ?ukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MV-algebra Free n is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0, 1] n . The maximal spectrum of Free n is canonically homeomorphic to [0, 1] n , and the automorphisms of the algebra are in 1–1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this article, we prove that the only probability measure on [0, 1] n which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic logic, it means that the only automorphism-invariant truth averaging process that detects pseudotrue propositions is the integral with respect to Lebesgue measure. 相似文献
18.
The Kopytov order for any algebra over a field is considered. Necessary and sufficient conditions for an algebra to be a linearly ordered algebra are presented. Some results concerning the properties of ideals of linearly ordered algebras are obtained. Some examples of algebras with the Kopytov order are described. The Kopytov order for these examples induces the order on other algebraic objects. The purpose of this paper is to investigate a generalization of the concept of prime radical to lattice-ordered algebras over partially ordered fields. Prime radicals of l-algebras over partially ordered and directed fields are described. Some results concerning the properties of the lower weakly solvable l-radical of l-algebras are obtained. Necessary and sufficient conditions for the l-prime radical of an l-algebra to be equal to the lower weakly solvable l-radical of the l-algebra are presented. 相似文献
19.
Lattice orders on the semigroup ring of a positive rooted monoid are constructed, and it is shown how to make the monoid ring into a lattice-ordered ring with squares positive in various ways. It is proved that under certain conditions these are all of the lattice orders that make the monoid ring into a lattice-ordered ring. In particular, all of the partial orders on the polynomial ring A[x] in one positive variable are determined for which the ring is not totally ordered but is a lattice-ordered ring with the property that the square of every element is positive. In the last section some basic properties of d-elements are considered, and they are used to characterize lattice-ordered division rings that are quadratic extensions of totally ordered division rings. 相似文献
20.
Franco Montagna 《Journal of Pure and Applied Algebra》2010,214(1):71-70
Our work proposes a new paradigm for the study of various classes of cancellative residuated lattices by viewing these structures as lattice-ordered groups with a suitable operator (a conucleus). One consequence of our approach is the categorical equivalence between the variety of cancellative commutative residuated lattices and the category of abelian lattice-ordered groups endowed with a conucleus whose image generates the underlying group of the lattice-ordered group. In addition, we extend our methods to obtain a categorical equivalence between -algebras and product algebras with a conucleus. Among the other results of the paper, we single out the introduction of a categorical framework for making precise the view that some of the most interesting algebras arising in algebraic logic are related to lattice-ordered groups. More specifically, we show that these algebras are subobjects and quotients of lattice-ordered groups in a “quantale like” category of algebras. 相似文献