On subtractive varieties II: General properties

Aglianò, Paolo; Ursini, Aldo

doi:10.1007/bf01234106

Cited by 35 publications

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“…If A is an algebra a semicongruence [2] of A is a reflexive subalgebra of A × A; Con(A) is the congruence lattice of A.…”

Section: Preliminariesmentioning

confidence: 99%

“…(1) implies (2), simply because the mapping is the composition of the hoomorphisms 0/ and F. Assume (2) and let I =0/q, J= 0/; then…”

Section: Proofmentioning

confidence: 99%

“…In the next sections we will study them, giving conditions under which they are normalizers, and consequences thereof. The behavior of the commutator of ideals ( [2], [15]) will be investigated in a different paper.…”

Section: Proofmentioning

confidence: 99%

“…The last section of the paper is devoted to examples of varieties with (strongly) normal ideals; this serves both to illuminate the entire subject and also to emphasize that the concept of an algebra with (strongly) normal ideals applies to a fairly broad spectrum of structures . We tried to make the present paper as self-contained as possible; the few results in [2], [15] to which we refer are easily proved if needed.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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On subtractive varieties III: From ideals to congruences

Aglianò¹,

Ursini²

1997

Algebra Universalis

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As a sequel to [2] and [15] we investigate ideal properties focusing on subtractive varieties. Here we probe the relations between congruences and ideals in subtractive varieties, in order to give some means to recover the congruence structure from the ideal structure. To do so we consider mainly two operators from the ideal lattice to the congruence lattice of a given algebra and we classify subtractive varieties according to various properties of these operators. In the last section several examples are discussed in details. IntroductionThe aim of this paper is to investigate the relation ''congruences-ideals'' in subtractive algebras. In a sense we want to see how short the algebras fall from being 0-regular (in which case the relation is the best possible: a lattice isomorphism), and also give a sensible way to control the set CON(I) = {q Con(A): 0/q =I} of the congruences associated to an ideal I, in order to ascertain something of the structure of the family {CON(I): I I(A)}.A variety V has normal ideals if for all A V every ideal of A is a congruence class; subtractive algebras do have normal ideals. In particular we consider the two natural mappings ( ) l and ( ) m which associate to any ideal I the least (resp. the greatest) congruence in CON(I). If V is a variety with normal ideals we set V m = {A/(0) m A : A V}; then algebraic properties of V and closure properties of V m turn out to be connected with logical properties of the assertional logic AL V (in the sense of Blok and Pigozzi [6]). For instance if V has normal ideals, then V is subtractive iff AL V is protoalgebraic iff V m is closed under subdirect products iff ( ) m is monotonic. V is ideal determined iff V m is a variety iff ( ) m is a homomorphism iff AL V is strongly algebraizable.

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“…If A is an algebra a semicongruence [2] of A is a reflexive subalgebra of A × A; Con(A) is the congruence lattice of A.…”

Section: Preliminariesmentioning

confidence: 99%

“…(1) implies (2), simply because the mapping is the composition of the hoomorphisms 0/ and F. Assume (2) and let I =0/q, J= 0/; then…”

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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