共查询到20条相似文献,搜索用时 15 毫秒
1.
We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher
commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order
and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras,
we can check in polynomial time whether two given polynomial terms induce the same function. 相似文献
2.
3.
Alex Citkin 《Algebra Universalis》2018,79(3):66
We prove that in the varieties where every compact congruence is a factor congruence and every nontrivial algebra contains a minimal subalgebra, a finitely presented algebra is projective if and only if it has every minimal algebra as its homomorphic image. Using this criterion of projectivity, we describe the primitive subquasivarieties of discriminator varieties that have a finite minimal algebra embedded in every nontrivial algebra from this variety. In particular, we describe the primitive quasivarieties of discriminator varieties of monadic Heyting algebras, Heyting algebras with regular involution, Heyting algebras with a dual pseudocomplement, and double-Heyting algebras. 相似文献
4.
The set \({{\mathrm{Quo}}}(\mathbf {A})\) of compatible quasiorders (reflexive and transitive relations) of an algebra \(\mathbf {A}\) forms a lattice under inclusion, and the lattice \({{\mathrm{Con}}}(\mathbf {A})\) of congruences of \(\mathbf {A}\) is a sublattice of \({{\mathrm{Quo}}}(\mathbf {A})\). We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice \(\mathbf {M}_3\). We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices. 相似文献
5.
The relationship between absolute retracts, injectives and equationally compact algebras in finitely generated congruence
distributive varieties with 1- element subalgebras is considered and several characterization theorems are proven. Amongst
others, we prove that the absolute retracts in such a variety are precisely the injectives in the amalgamation class and that
every equationally compact reduced power of a finite absolute retract is an absolute retract. We also show that any elementary
amalgamation class is Horn if and only if it is closed under finite direct products.
The second author's work was supported by grants from the South African Council for Scientific and Industrial Research and
the University of Cape Town Research Committee. 相似文献
6.
7.
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
8.
For varieties of algebras, we present the property of having "definable principal subcongruences" (DPSC), generalizing the
concept of having definable principal congruences. It is shown that if a locally finite variety V of finite type has DPSC, then V has a finite equational basis if and only if its class of subdirectly irreducible members is finitely axiomatizable. As an
application, we prove that if A is a finite algebra of finite type whose variety V(A) is congruence distributive, then V(A) has DPSC. Thus we obtain a new proof of the finite basis theorem for such varieties. In contrast, it is shown that the group
variety V(S
3
) does not have DPSC.
Received May 9 2000; accepted in final form April 26, 2001. 相似文献
9.
PaweŁ M. Idziak 《Algebra Universalis》1989,26(2):247-256
This paper is a continuation of [3]. Congruence permutability is shown to be a necessary condition for a locally finite congruence distributive variety to have a decidable first order theory of its finite algebras. This is a positive answer to Problem 6 of S. Burns and H. P. Sankappanavar [2]. Moreover this allows us to give a full characterization of finitely generated congruence distributive varieties of finite type with decidable first order theories of their finite members.Presented by Stanley Burris. 相似文献
10.
Two varieties generated by finite algebras and , respectively are categorically equivalent under an equivalence functor which takes to iff the algebras of invariant relations of the clones of all term operations of and are isomorphic. In this paper we will prove this theorem and will give several applications.
Received September 10, 1999; accepted in final form October 16, 2000. 相似文献
11.
PaweŁ M. Idziak 《Algebra Universalis》1989,26(2):234-246
The aim of this paper is to prove that every congruence distributive variety containing a finite subdirectly irreducible algebra whose congruences are not linearly ordered has an undecidable first order theory of its finite members. This fills a gap which kept us from the full characterization of the finitely generated, arithmetical varieties (of finite type) having a decidable first order theory of their finite members. Progress on finding this characterization was made in the papers [14] and [15].Presented by Stanley Burris. 相似文献
12.
An algebra is called affine complete if all its compatible (i.e. congruence-preserving) functions are polynomial functions.
In this paper we characterize affine complete members in the variety of Kleene algebras. We also characterize local polynomial
functions of Kleene algebras and use this result to describe locally affine complete Kleene algebras.
Received December 20, 1996; accepted in final form March 24, 1997. 相似文献
13.
Matthew Moore 《Algebra Universalis》2016,75(2):221-230
An early result in the theory of Natural Dualities is that an algebra with a near unanimity (NU) term is dualizable. A converse to this is also true: if \({\mathcal{V}(\mathbb{A})}\) is congruence distributive and \({\mathbb{A}}\) is dualizable, then \({\mathbb{A}}\) has an NU term. An important generalization of the NU term for congruence distributive varieties is the cube term for congruence modular (CM) varieties, and it has been thought that a similar characterization of dualizability for algebras in a CM variety would also hold. We prove that if \({\mathbb{A}}\) omits tame congruence types 1 and 5 (all locally finite CM varieties omit these types) and is dualizable, then \({\mathbb{A}}\) has a cube term. 相似文献
14.
Miroslav Ploščica 《Algebra Universalis》2003,49(1):1-12
15.
An algebra A is endoprimal if, for all the only maps from A
k
to A which preserve the endomorphisms of A are its term functions. One method for finding finite endoprimal algebras is via the theory of natural dualities since an
endodualisable algebra is necessarily endoprimal. General results on endoprimality and endodualisability are proved and then
applied to the varieties of sets, vector spaces, distributive lattices, Boolean algebras, Stone algebras, Heyting algebras,
semilattices and abelian groups. In many classes the finite endoprimal algebras turn out to be endodualisable. We show that
this fails in general by proving that , regarded as either a bounded semilattice or upper-bounded semilattice is dualisable, endoprimal but not endodualisable.
Received May 16, 1997; accepted in final form November 6, 1997. 相似文献
16.
Let P be a property of topological spaces. Let [P] be the class of all varieties having the property that any topological algebra in has underlying space satisfying property P. We show that if P is preserved by finite products, and if is preserved by ultraproducts, then [P] is a class of varieties that is definable by a Maltsev condition.?The property that all T
0
topological algebras in are j-step Hausdor. (Hj) is preserved by finite products, and its negation is preserved by ultraproducts. We partially characterize
the Maltsev condition associated to by showing that this topological implication holds in every (2j + 1)-permutable variety, but not in every (2j + 2)-permutable variety.?Finally, we show that the topological implication holds in every k-permutable, congruence modular variety.
Received March 1, 2000; accepted in final form October 18, 2001. 相似文献
17.
Ralph McKenzie 《Algebra Universalis》1987,24(3):224-250
In this paper it is proved that a variety generated by a finite algebraic system with finitely many operations is finitely axiomatizable, provided that the variety is congruence modular and residually small. This result is an extension to congruence modular varieties of a well known theorem for congruence distributive varieties, due to K. A. Baker. Also, under somewhat less restrictive hypotheses, (which are satisfied by finite groups and rings) it is proved that a finite algebraic system belongs to a finitely axiomatizable locally finite variety.Research supported by National Science Foundation Grant No. DMS-8302295.Presented by George Gratzer. 相似文献
18.
We propound an approach through which techniques of the theory of quasivarieties of predicate systems are brought to bear
on partial algebras. For every partial algebra A, two predicate representations are treated. The first is the graph of A whose
basic operations are graphs of the basic operations on A. The second representation results from the graph of A by adding
domains of the operations on A to its basic relations. Studying partial algebras from various perspectives makes it necessary
to deal with different equality semantics. Here we present a general definition of semantics that stretches over such instances
as weak semantics, Evans’ semantics. Kleene semantics, and strong semantics. On a set of all semantics, the preorder is induced
in increasing “force,” and it is proved that certain of the properties of varieties of partial algebras in a given semantics
are individuated by the position it takes in that set. We argue that every variety of partial algebras, in any semantics,
is in correspondence with a Horn class of predicate systems which admits a generation operator and is closed under direct
limits and retracts. For such classes we prove analogs of the Birkhoff theorem on subdirect decompositions and of the Taylor
theorem on residual smallness. Therefore, these are also applicable to varieties of partial algebras in arbitrary semantics.
Supported through the RF State Committee of Higher Education (1998 project), jointly by RFFR and DFG grants Nos. 96-01-00097
and 436113/2670, and also through FP “Integration” project No. 274.
Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 23–46, January–February, 2000. 相似文献
19.
Keith A. Kearnes 《Algebra Universalis》2000,44(1-2):39-45
We show that, up to term equivalence, the only minimal idempotent varieties that are not congruence modular are the variety
of sets and the variety of semilattices. From this it follows that a minimal idempotent variety that is not congruence distributive
is term equivalent to the variety of sets, the variety of semilattices, or a variety of affine modules over a simple ring.
Received March 29, 1999; accepted in final form February 8, 2000. 相似文献
20.
An algebraically expandable class is a class of similar algebras axiomatizable by sentences of the form ${(\forall\exists ! \bigwedge eq)}$ . The problem investigated in this work is that of finding all algebraically expandable classes within a given variety. A complete solution is presented for a number of varieties, including the classes of Boolean algebras, Stone algebras, semilattices, distributive lattices and generalized Kleene algebras. We also study the problem for the case of discriminator varieties, where we prove that there is a lattice isomorphism between the lattice of all algebraically expandable classes of the variety and a certain lattice of subclasses of the simple members of the variety. Finally this connection is applied to calculating the algebraically expandable subclasses of the varieties of monadic algebras and P-algebras. 相似文献