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1.
We show that a locally finite variety satisfies a nontrivial congruence identity if and only if it satisfies an idempotent Mal'tsev condition that fails in the variety of semilattices. Received January 27, 1999; accepted in final form June 11, 1999.  相似文献   

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In this paper we describe a one-variable Mal'cev-like condition satisfied by any locally finite minimal variety. We prove that a locally finite variety is minimal if and only if it satisfies this Mal'cev-like condition and it is generated by a strictly simple algebra which is nonabelian or has a trivial subalgebra. Our arguments show that the strictly simple generator of a minimal locally finite variety is unique, it is projective and it embeds into every member of the variety. We give a new proof of the structure theorem for strictly simple abelian algebras that generate minimal varieties.

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The Loewy rank of a modular latticeL of finite height is defined as the leastn for which there exista 0=0t, < ... r=1 inL such that each interval I[ai, ai+1] is a complemented lattice. In this paper, a generalized notion of Loewy rank is applied to obtain new results in the commutator theory of locally finite congruence modular varieties. LetV be a finitely generated congruence modular variety. We prove that every algebra inV has a largest nilpotent congruence and a largest solvable congruence. Moreover, there exist first order formulas which define these special congruences in every algebra ofV.  相似文献   

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In an attempt to describe the partially ordered monoid of operators generated by the operators H (homomorphic images), S (subalgebras), \({P_{\rm f}}\) (filtered products) for the variety \({\mathcal{R}_{\rm c}}\) of commutative rings, several results about congruence permutable varieties have been discovered.  相似文献   

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Locally finite, congruence meet-semidistributive varieties have been characterized by numerous Mal’cev conditions and, recently, by two strong Mal’cev conditions. We provide three new strong Mal’cev characterizations and a new Mal’cev characterization each of which improves the known ones in some way.  相似文献   

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We define a weak version of EDPC (equationally definable principal congruences), called EDPC*, that is shown to be preserved under varietal closure in congruence permutable varieties. We show that if is a congruence permutable variety generated by a class then has EDPC iff has EDPC* iff has EDPC*. An equational condition is given which, if satisfied by implies that has the CEP (congruence extension property). Similar results are proved for ideal determined varieties. These results are applied to the variety of residuated lattices, with examples.Received January 15, 2004; accepted in final form October 8, 2004.  相似文献   

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A locally finite graph G with no isolated vertices is vertex-transitive if and only if all its vertex-deleted subgraphs G-v are isomorphic.  相似文献   

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We introduce new sufficient conditions for a finite algebraU to possess a finite basis of identities. The conditions are that the variety generated byU possess essentially only finitely many subdirectly irreducible algebras, and have definable principal congruences. Both conditions are satisfied if this variety is directly representable by a finite set of finite algebras. One task of the paper is to show that virtually no lattice varieties possess definable principal congruences. However, the main purpose of the paper is to apply the new criterion in proving that every para primal variety (congruence permutable variety generated by finitely many para primal algebras) is finitely axiomatizable. The paper also contains a completely new approach to the structure theory of para primal varieties which complements and extends somewhat the recent work of Clark and Krauss.  相似文献   

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We show that a residually finite, congruence meet-semidistributive variety of finite type is residually for some finite . This solves Pixley's problem and a special case of the restricted Quackenbush problem.

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In some recent papers, the concept of a Q-independent sequence of finite lattices was utilized. We investigate this concept in universal algebras and apply it to positive universal classes in locally finite varieties, with emphasis on semilattices, lattices, and their expansions.  相似文献   

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We provide several conditions that, among locally finite varieties, characterize congruence meet-semidistributivity and we use these conditions to give a new proof of a finite basis theorem published by Baker, McNulty, and Wang in 2004. This finite basis theorem extends Willard’s Finite Basis Theorem.  相似文献   

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No Abstract. Received July 8, 1999; accepted in final form January 31, 2001.  相似文献   

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In this note, we characterize finite modules locally of finite injective dimension over commutative Noetherian rings in terms of vanishing of Ext modules.

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18.
On permutable subgroups of finite groups   总被引:6,自引:0,他引:6  
Let \frak Z \frak Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, \frak Z \frak Z contains exactly one and only one Sylow p-subgroup of G. A subgroup H of a finite group G is said to be \frak Z \frak Z -permutable if H permutes with every member of \frak Z \frak Z . The purpose here is to study the influence of \frak Z \frak Z -permutability of some subgroups on the structure of finite groups. Some recent results are generalized.  相似文献   

19.
We prove that if a finite connected poset admits an order-preserving Taylor operation, then all of its homotopy groups are trivial. We use this to give new characterisations of locally finite varieties omitting type 1 in terms of the posets (or equivalently, finite topological spaces) in the variety. Similar variants of other omitting-type theorems are presented. We give several examples of posets that admit various types of Taylor operations; in particular, we exhibit a topological space which is not an H-space but is compatible with a set of non-trivial identities, answering a question of W. Taylor.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived September 18, 2002; accepted in final form March 19, 2003.  相似文献   

20.
Some of the results obtained by A. Pinus for congruence distributive varieties are generalized to the case of congruence-nodular varieties. Most attention is paid to congruence-modular varieties containing a subdirectly indecomposable algebra with non-Abelian monolith. Supported by RFFR grant No. 93-011-1520. Translated fromAlgebra i Logika, Vol. 35, No. 1, pp. 79–87, January–February, 1996.  相似文献   

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