Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 290-321 |
| Seitenumfang | 32 |
| Fachzeitschrift | Algebra universalis |
| Jahrgang | 25 |
| Ausgabenummer | 1 |
| Publikationsstatus | Veröffentlicht - Dez. 1988 |
Abstract
By a result of Pigozzi and Kogalovskii, every algebraic lattice L having a completely join -irreducible top element can be represented as the lattice L(Σ) of equational theories extending some fixed theory Σ. Conversely, strengthening a recent result due to Lampe, we show that such a representation L=L(Σ) forces L to satisfy the following condition: if the top element of L is the join of a nonempty subset B of L then there are elements b..., ε B such that a=(... (((b1 ∧a) ∨ b2) ∧a) ... ∨ bn) ∧a for all a ε L. In presence of modularity, this equation reduces to the identity a=(a ∧ b1) ∨ ... ∨ (a ∧ bn). Motivated by these facts, we study several weak forms of distributive laws in arbitrary lattices and related types of prime elements. The main tool for applications to universal algebra is a generalized version of Lampe's Zipper Lemma.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Algebra und Zahlentheorie
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in: Algebra universalis, Jahrgang 25, Nr. 1, 12.1988, S. 290-321.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Weak distributive laws and their role in lattices of congruences and equational theories
AU - Erné, Marcel
PY - 1988/12
Y1 - 1988/12
N2 - By a result of Pigozzi and Kogalovskii, every algebraic lattice L having a completely join -irreducible top element can be represented as the lattice L(Σ) of equational theories extending some fixed theory Σ. Conversely, strengthening a recent result due to Lampe, we show that such a representation L=L(Σ) forces L to satisfy the following condition: if the top element of L is the join of a nonempty subset B of L then there are elements b..., ε B such that a=(... (((b1 ∧a) ∨ b2) ∧a) ... ∨ bn) ∧a for all a ε L. In presence of modularity, this equation reduces to the identity a=(a ∧ b1) ∨ ... ∨ (a ∧ bn). Motivated by these facts, we study several weak forms of distributive laws in arbitrary lattices and related types of prime elements. The main tool for applications to universal algebra is a generalized version of Lampe's Zipper Lemma.
AB - By a result of Pigozzi and Kogalovskii, every algebraic lattice L having a completely join -irreducible top element can be represented as the lattice L(Σ) of equational theories extending some fixed theory Σ. Conversely, strengthening a recent result due to Lampe, we show that such a representation L=L(Σ) forces L to satisfy the following condition: if the top element of L is the join of a nonempty subset B of L then there are elements b..., ε B such that a=(... (((b1 ∧a) ∨ b2) ∧a) ... ∨ bn) ∧a for all a ε L. In presence of modularity, this equation reduces to the identity a=(a ∧ b1) ∨ ... ∨ (a ∧ bn). Motivated by these facts, we study several weak forms of distributive laws in arbitrary lattices and related types of prime elements. The main tool for applications to universal algebra is a generalized version of Lampe's Zipper Lemma.
UR - http://www.scopus.com/inward/record.url?scp=34250097696&partnerID=8YFLogxK
U2 - 10.1007/BF01229979
DO - 10.1007/BF01229979
M3 - Article
AN - SCOPUS:34250097696
VL - 25
SP - 290
EP - 321
JO - Algebra universalis
JF - Algebra universalis
SN - 0002-5240
IS - 1
ER -