On the semigroup structure of cyclic left distributive algebras
type: Item
instance of: scholarly article
title: On the semigroup structure of cyclic left distributive algebras
zbMATH Open document ID: 0828.20070
DOI: 10.1007/BF02573617
author: Aleš Drápal
published in: Semigroup Forum
publication date: 1995-12-18T00:00:00Z
full work available at URL: https://eudml.org/doc/135397
review text: An algebra \(G(\circ, *)\) is called left distributive if it satisfies the identities \((a \circ b) \circ c = a \circ (b \circ c)\), \((a \circ b) * c = a * (b * c)\), \(a \circ b = (a * b) \circ a\), \(a * (b \circ c) = (a * b) \circ (a * c)\). It follows that the operation \(*\) is left distributive: \(a * (b * c) = (a * b) * (a * c)\). For every \(n > 0\) there exists just one left distributive operation \(*\) on the set \(P_n = \{0, 1, \dots, 2^n - 1\}\) such that \(a * 1 = (a + 1) \text{ mod } 2^n\); the operation \(\circ\) may be defined on \(P_n\) by \((a \circ b) * 1 = a * (b * 1)\). Idempotents, regular elements and generators of the semigroup \((P_n, \circ)\) are described; e.g. \((P_n, \circ)\) is generated by the set \(R_n\) of rare elements, i.e. elements \(a \in P_n\) such that \(a = b * c\) implies \(b= 0\) or \(b * 1 = a\) for any \(b, c \in P_n\).
Mathematics Subject Classification ID: 20N02
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 08A02
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 784128
zbMATH Keywords: idempotents
zbMATH Keywords: left distributive operation
zbMATH Keywords: regular elements
zbMATH Keywords: generators
zbMATH Keywords: rare elements
reviewed by: Jaak Henno
MaRDI profile type: Publication
cites work: Q3848243
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cites work: Q3981500
OpenAlex ID: W2005159206
description: scientific article; zbMATH DE number 784128