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1.
In this paper we will generalize the representation theory developed for finite Tarski algebras given in [7]. We will introduce
the notion of Tarski space as a generalization of the notion of dense Tarski set, and we will prove that the category of Tarski
algebras with semi-homomorphisms is dually equivalent to the category of Tarski spaces with certain closed relations, called
T-relations. By these results we will obtain that the algebraic category of Tarski algebras is dually equivalent to the category
of Tarski spaces with certain partial functions. We will apply these results to give a topological characterization of the
subalgebras.
Received August 21, 2005; accepted in final form December 5, 2006. 相似文献
2.
Elias David 《Algebra Universalis》1993,30(2):221-233
There is a canonical imbedding of a poset into a complete Boolean lattice and hence into a Boolean lattice. This gives it a representation as a collection of clopen sets of a Boolean space. There are reflective functions from a category of distributive posets to the subcategories of distributive and Boolean lattices and consequently a topological dual equivalence that extends the Stone duality of Boolean lattices.Presented by B. Jonsson. 相似文献
3.
On amalgamation of reducts of polyadic algebras 总被引:3,自引:0,他引:3
Tarek Sayed Ahmed 《Algebra Universalis》2004,51(4):301-359
Following research initiated by Tarski, Craig and Németi, and further pursued
by Sain and others, we show that for certain subsets G of
w,
G polyadic algebras have
the strong amalgamation property. G polyadic algebras are obtained by restricting the
(similarity type and) axiomatization of -dimensional polyadic algebras to finite quantifiers
and substitutions in G. Using algebraic logic, we infer that some theorems of Beth, Craig
and Robinson hold for certain proper extensions of first order logic (without equality). 相似文献
4.
P. Jipsen 《Annals of Pure and Applied Logic》2009,161(2):228-234
It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FLw-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras are represented as Esakia products of simple n-potent MV-algebras. 相似文献
5.
Saharon Shelah 《Algebra Universalis》2005,54(1):91-96
We show that if μ is a compact cardinal then the depth of ultraproducts of less than μ many Boolean algebras is at most μ
plus the ultraproduct of the depths of those Boolean algebras.
Received May 18, 2004; accepted in final form December 9, 2004. 相似文献
6.
《Quaestiones Mathematicae》2013,36(1-4):69-94
ABSTRACT This paper generalizes the concept of a power alge bra to that of a power structure, and gives three application of power structures to logic. 相似文献
7.
Injectives in several classes of structures associated with logic are characterized.
Among the classes considered are residuated lattices, MTL-algebras, IMTL-algebras, BL-algebras,
NM-algebras and bounded hoops. 相似文献
8.
The poset product construction is used to derive embedding theorems for several classes of generalized basic logic algebras (GBL-algebras). In particular it is shown that every n-potent GBL-algebra is embedded in a poset product of finite n-potent MV-chains, and every normal GBL-algebra is embedded in a poset product of totally ordered GMV-algebras. Representable normal GBL-algebras have poset product embeddings where the poset is a root system. We also give a Conrad-Harvey-Holland-style embedding theorem for commutative GBL-algebras, where the poset factors are the real numbers extended with −∞. Finally, an explicit construction of a generic commutative GBL-algebra is given, and it is shown that every normal GBL-algebra embeds in the conucleus image of a GMV-algebra. 相似文献
9.
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
10.
In this paper we present several results about local MV-algebras, extending existing results given for MV-chains. The role
of local MV-algebras in sheaf representation and weak boolean product is stressed and the relationship of local MV-algebras
with varieties of MV-algebras is analyzed.
Presented by S. Pulmannova.
Received November 11, 2005; accepted in final form December 20, 2005. 相似文献
11.
We introduce and investigate topo-canonical completions of closure algebras and Heyting algebras. We develop a duality theory
that is an alternative to Esakia’s duality, describe duals of topo-canonical completions in terms of the Salbany and Banaschewski
compactifications, and characterize topo-canonical varieties of closure algebras and Heyting algebras. Consequently, we show
that ideal completions preserve no identities of Heyting algebras. We also characterize definable classes of topological spaces.
Received January 20, 2006; accepted in final form September 12, 2006. 相似文献
12.
13.
Pseudoeffect (PE-) algebras are partial algebras differing from effect algebras
in that they need not satisfy the commutativity assumption. PE-algebras typically arise
from intervals of po-groups; this applies in particular to all those which satisfy a certain
Riesz property.In this paper, we discuss the property of archimedeanness for PE-algebras on the one
hand and for po-groups on the other hand. We prove that under the assumption of suphomogeneity,
archimedeanness holds for a PE-algebra with the Riesz property if and only
if it holds for its representing group. The algebra is in that case commutative.
This result is established by using the technique of MacNeille completion. We give
the exact condition for this completion to exist, and we clearly exhibit the role played by
archimedeanness and by sup-homogeneity. 相似文献
14.
Thomas Vetterlein 《Algebra Universalis》2008,58(2):129-143
Weak effect algebras are based on a commutative, associative and cancellative partial addition; they are moreover endowed
with a partial order which is compatible with the addition, but in general not determined by it. Every BL-algebra, i.e. the
Lindenbaum algebra of a theory of Basic Logic, gives rise to a weak effect algebra; to this end, the monoidal operation is
restricted to a partial cancellative operation.
We examine in this paper BL-effect algebras, a subclass of the weak effect algebras which properly contains all weak effect
algebras arising from BL-algebras. We describe the structure of BL-effect algebras in detail. We thus generalise the well-known
structure theory of BL-algebras.
Namely, we show that BL-effect algebras are subdirect products of linearly ordered ones and that linearly ordered BL-effect
algebras are ordinal sums of generalised effect algebras. The latter are representable by means of linearly ordered groups.
This research was partially supported by the German Science Foundation (DFG) as part of the Collaborative Research Center
“Computational Intelligence” (SFB 531). 相似文献
15.
We consider Boolean algebras constructed from pseudo-trees in various ways and make comments about related classes of Boolean algebras. 相似文献
16.
C. Jayaram 《Algebra Universalis》2006,55(2-3):297-303
In this paper we establish several equivalent conditions for an algebraic lattice to be a finite Boolean algebra.
This paper is dedicated to Walter Taylor.
Received February 11, 2005; accepted in final form October 9, 2005. 相似文献
17.
R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF. 相似文献
18.
A finite axiom set for the identity-free equations valid in relation algebras is given. This is a simplification of the one given by Jónsson, and confirms a conjecture of Tarski. An axiom set for the identity-free equations valid in the representable relation algebras is given, too. We show that in the class of representable relation algebras, both the operation of taking converse and the identity constant are finitely axiomatizable (over the rest of the operations).Dedicated to the memory of Alan DayPresented by J. Sichler. 相似文献
19.
20.
An effect algebra is a partial algebra modeled on the standard effect algebra of positive self-adjoint operators dominated by the identity on a Hilbert space. Every effect algebra is partially ordered in a natural way, as suggested by the partial order on the standard effect algebra. An effect algebra is said to be distributive if, as a poset, it forms a distributive lattice. We define and study the center of an effect algebra, relate it to cartesian-product factorizations, determine the center of the standard effect algebra, and characterize all finite distributive effect algebras as products of chains and diamonds. 相似文献