Reductive modes
type: Item
instance of: scholarly article
title: Reductive modes
zbMATH Open document ID: 0924.08001
DOI: 10.1023/A:1004612003441
published in: Periodica Mathematica Hungarica
publication date: 1999-10-31T00:00:00Z
review text: A mode is an idempotent and entropic algebra, i.e., every singleton is a subalgebra and each operation is a homomorphism. A mode is \(n\)-step left reductive if it has a binary term and satisfies the identity: \[ x_1\cdot (x_2\cdot \dots(x_n\cdot y)\dots)=x_1\cdot(x_2\cdot\dots(x_n\cdot z)\dots). \] By \(R_n\) we denote the variety of \(n\)-step left reductive modes, the variety \(R_n'\) of \(n\)-step right reductive modes is defined symmetrically. Main results: Theorem 1.6. Let \(k<n\) be integers and \(\circ\) denote the Mal'tsev product of varieties. Then \(R_n=R_k\circ R_{n-k}\). Corollary: Every variety of \(n\)-step left reductive binary modes is the \(n\)th Mal'tsev power of the variety of left-zero bands. Theorem 2.3. The join \(R_m\vee R'_n\) is independent and coincides with \(R_m\circ R'_n\). The authors describe explicitly an equational characterization of \(R_m\vee R'_n\).
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 08A05
Mathematics Subject Classification ID: 08B25
Mathematics Subject Classification ID: 03C05
zbMATH DE Number: 1354873
zbMATH Keywords: modes
zbMATH Keywords: Mal'tsev product
zbMATH Keywords: independent varieties
zbMATH Keywords: sums of algebras
zbMATH Keywords: identities
zbMATH Keywords: idempotent and entropic algebra
zbMATH Keywords: variety of \(n\)-step left reductive modes
zbMATH Keywords: Mal'tsev power
author: Agata Pilitowska
author: Anna Romanowska
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1354873