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1.
An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form avariety SCIL, which is not locally finite, but it is proved that SCIL has the finite embeddability property (FEP). More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains. 相似文献
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This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which either treated fewer subsignatures or focussed on the conservation of theorems only. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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M. V. Volkov 《Algebra Universalis》2001,46(1-2):97-103
No Abstract. Received January 2, 2000; accepted in final form August 28, 2000. 相似文献
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Adjoint triples and pairs are basic operators used in several domains, since they increase the flexibility in the framework in which they are considered. This paper introduces multi-adjoint algebras and several properties; also, we will show that an adjoint triple and its “dual” cannot be considered in the same framework.Moreover, a comparison among general algebraic structures used in different frameworks, which reduce the considered mathematical requirements, such as the implicative extended-order algebras, implicative structures, the residuated algebras given by sup-preserving aggregations and the conjunctive algebras given by semi-uninorms and u-norms, is presented. This comparison shows that multi-adjoint algebras generalize these structures in domains which require residuated implications, such as in formal concept analysis, fuzzy rough sets, fuzzy relation equations and fuzzy logic. 相似文献
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Wiesław Dziobiak 《Algebra Universalis》1991,28(3):303-323
In [21], D. Pigozzi has proved in a non-constructive way that every relatively congruence distributive quasivariety of finite type generated by a finite set of finite algebras is finitely axiomatizable. In this paper we show that the non-constructive parts of Pigozzi's argument can be replaced by constructive ones. As a result we obtain a method of constructing a finite set of quasi-equational axioms for each relatively congruence distributive quasivariety generated by a given finite set of finite algebras of finite type. The method can also be applied to finitely generated congruence distributive varieties.Presented by Joel Berman. 相似文献
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We classified in Bisch and Jones (Duke Math. J. 101 (2000) 41) all spherical C∗-planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of N′∩M2 is ?12. We showed that they are given by the Fuss-Catalan systems discovered in Bisch and Jones (Invent. Math. 128 (1997) 89) and one exceptional planar algebra. In the present paper, we extend these results and show that there is only one spherical C∗-planar algebra generated by a single non-trivial 2-box if the dimension of N′∩M2 is 13. It is given by the standard invariant of the crossed product subfactor , where D5 denotes the dihedral group with 10 elements. 相似文献
9.
Annika M. Wille 《Archiv der Mathematik》2006,87(6):546-553
It will be shown that the word problem is undecidable for involutive residuated lattices, for finite involutive residuated
lattices and certain related structures like residuated lattices. The proof relies on the fact that the monoid reduct of a
group can be embedded as a monoid into a distributive involutive residuated lattice. Thus, results about groups by P. S. Novikov
and W. W. Boone and about finite groups by A. M. Slobodskoi can be used. Furthermore, for any non-trivial lattice variety
, the word problem is undecidable for those involutive residuated lattices and finite involutive residuated lattices whose
lattice reducts belong to
. In particular, the word problem is undecidable for modular and distributive involutive residuated lattices.
The author would like to thank the Deutsche Telekom Stiftung for financial support.
Received: 10 November 2005 相似文献
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Categorical quasivarieties revisited 总被引:1,自引:0,他引:1
Ralph Mckenzie 《Algebra Universalis》1984,19(3):273-303
Quasivarieties (and varieties) which are categorical in some power not less than the power of their language have been completely characterized by S. Givant [6], [7] and independently by E. A. Palyutin [11], [12]. These classes fall into two radically different families. A class in the first family is derived from the class of permutational representations of a group. Its members are [n]-th powers of algebras whose operations are unary, for some fixed positive integern. A class in the second family consists of affine algebras. Its members are polynomially equivalent (but not usually definitionally equivalent) to modules over a ring which is isomorphic to the ring ofn-by-n matrices with entries in a division ring.The general results are faithfully represented in the family of-categorical quasivarieties of countable type. Each of them is generated by a finite algebra, and the results can be viewed as very interesting facts about finite algebras and the classes they generate. In this paper, we offer simple new characterizations of-categorical quasivarieties and varieties of countable type. Our arguments are distinguished by the absence of any sophisticated model theory. In the beginning we use some very basic model theory, but after that we find that combinatorial reasoning about finite sets and elementary algebraic arguments, combined with two classical theorems describing the structure of finite simple rings and their modules, suffice to derive the results. Theorems 3.1 and 4.12 combine to give the characterization of-categorical quasivarieties. Theorems 3.2 and 4.13 combine to give the characterization of-categorical varieties.The heart of this paper is §2. There we prove that a nontrivial algebra of least cardinality in an-categorical quasivariety (which must generate the class) is a finite tame algebra. Tameness is the principal tool used in a relatively quick and painless proof that the generating algebra must be affine or an [n]-th power of a unary algebra. The concept of a tame algebra was introduced in [9] where we proved, among other things, that finite simple algebras are tame. When we had gained some experience with this concept, it became clear to us that the arguments in this present paper should exist (and it didn't take long to find them).The author thanks the referee for a thoughtful critique of the first submitted version of this paper.Research supported by United States National Science Foundation grant MCS 8103455.Presented by W. Taylor. 相似文献
13.
We investigate a construction of an integral residuated lattice starting from an integral residuated lattice and two sets with an injective mapping from one set into the second one. The resulting algebra has a shape of a Chinese cascade kite, therefore, we call this algebra simply a kite. We describe subdirectly irreducible kites and we classify them. We show that the variety of integral residuated lattices generated by kites is generated by all finite-dimensional kites. In particular, we describe some homomorphisms among kites. 相似文献
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The join semi-lattice of faces of an n-cube has a rich
structure. In considering generalizations of these structures we are led to looking at interval
algebras constructed using Boolean filters. We look at the structure of these algebras and
their automorphism groups.Dedicated to the memory of Gian-Carlo Rota 相似文献
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Yu. E. Shishmarev 《Algebra and Logic》1979,18(5):389-400
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Xiaohong Zhang 《Mathematica Slovaca》2013,63(3):397-410
The aim of the paper is to investigate the relationship between BCC-algebras and residuated partially-ordered groupoids. We prove that an integral residuated partially-ordered groupoid is an integral residuated pomonoid if and only if it is a double BCC-algebra. Moreover, we introduce the notion of weakly integral residuated pomonoid, and give a characterization by the notion of pseudo-BCI algebra. Finally, we give a method to construct a weakly integral residuated pomonoid (pseudo-BCI algebra) from any bounded pseudo-BCK algebra with pseudo product and any group. 相似文献
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Cancellative residuated lattices are natural generalizations of lattice-ordered
groups (
-groups).
Although cancellative monoids are defined by quasi-equations, the class
of cancellative residuated lattices is a variety.
We prove that there are only two
commutative subvarieties of
that cover the trivial variety, namely the varieties
generated by the integers and the negative integers (with zero). We also construct examples
showing that in contrast to
-groups, the lattice reducts of cancellative residuated lattices
need not be distributive. In fact we prove that every lattice can be embedded in the
lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an
order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of
.We define generalized MV-algebras and generalized BL-algebras and prove that the
cancellative integral members of these varieties are precisely the negative cones of
-groups, hence the latter form a variety, denoted by
. Furthermore we prove that the map that sends a subvariety of
-groups to the corresponding class of negative cones is a lattice
isomorphism from the lattice of subvarieties of
to the lattice of subvarieties of
.
Finally, we show how to translate equational bases between corresponding subvarieties, and
briefly discuss these results in the context of R. McKenzies characterization of categorically
equivalent varieties. 相似文献