On the varieties generated by ai-semirings of order two.
type: Item
instance of: scholarly article
title: On the varieties generated by ai-semirings of order two.
zbMATH Open document ID: 1347.16052
author: Yong Shao
author: Miaomiao Ren
published in: Semigroup Forum
publication date: 2015-12-09T00:00:00Z
review text: There are six non-isomorphic semirings of order two with idempotent addition. The variety \(\mathbf S_{\mathbf 2}\) generated by these semirings is finitely based by the identities \(xyzt\sim xzyt\), \((xy)^2\sim xy\) and \(x+yz\sim x+yz+xz+yx\). The authors show that the lattice \(L(\mathbf S_{\mathbf 2})\) of all subvarieties of \(\mathbf S_{\mathbf 2}\) is a Boolean algebra of order \(2^6\) and present a finite identity basis for each subvariety. This means that \(\mathbf S_{\mathbf 2}\) is hereditarily finitely based.
reviewed by: Udo Hebisch
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 6518146
zbMATH Keywords: varieties of semirings
zbMATH Keywords: additively idempotent semirings
zbMATH Keywords: bases of identities
zbMATH Keywords: hereditarily finitely based varieties
zbMATH Keywords: lattices of varieties
MaRDI profile type: Publication
full work available at URL: s00233-014-9667-z
OpenAlex ID: W1997962109
cites work: Q3934450
cites work: A NONFINITELY BASED FINITE SEMIRING
cites work: The finite basis problem for endomorphism semirings of finite semilattices with zero.
cites work: A class of inherently nonfinitely based semirings
cites work: A remark on nonfinitely based semirings.
cites work: Congruence-simple semirings.
cites work: Varieties generated by ordered bands. I.
cites work: Q4229053
cites work: Identities satisfied by a finite ring
cites work: On varieties of semilattice-ordered semigroups.
cites work: Varieties of associative rings. I
cites work: Identities in Two-Valued Calculi
cites work: Equational Bases for Lattice Theories.
cites work: Identical relations in finite groups
cites work: Varieties generated by ordered bands. II.
cites work: Weak commutativity in idempotent semirings.
cites work: The lattice of idempotent distributive semiring varieties
cites work: Bases for equational theories of semi-groups
cites work: Minimal varieties of semirings
cites work: Q5239598
cites work: Q3671238
cites work: \(\mathcal L\)-subvarieties of the variety of idempotent semirings.
cites work: Q4537209
DOI: 10.1007/S00233-014-9667-Z
Recommended article: Q4980366
Recommended article: Q3180899
Recommended article: THE VARIETY GENERATED BY AN AI-SEMIRING OF ORDER THREE
Recommended article: Q5196064
Recommended article: On some varieties of ai-semirings satisfying \(x^{p+1} \approx x\)
Recommended article: On a hereditarily finitely based ai-semiring variety
Recommended article: On varieties of semilattice-ordered semigroups.
Recommended article: Varieties of Burnside ai-semirings satisfying xn ≈ x
Recommended article: On the varieties of ai-semirings satisfying \({x^{3}\approx x}\)
Recommended article: The lattice of ai-semiring varieties satisfying \(x^n \approx x\) and \(xy \approx yx\)
description: scientific article; zbMATH DE number 6518146