On finite lattices which are embeddable in subsemigroup lattices
type: Item
instance of: scholarly article
title: On finite lattices which are embeddable in subsemigroup lattices
zbMATH Open document ID: 0797.20052
DOI: 10.1007/BF02573581
published in: Semigroup Forum
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/135245
review text: A lattice \(L\) is called lower bounded, if there exists a homomorphism \(f\) of a free lattice onto \(L\) such that for each \(a \in L\) the set \(f^{- 1}(a)\) has a least element. The free commutative nilsemigroup of index two of finite rank \(n\) is denoted by \(T_ n\) and the free semilattice of finite rank \(n\) is denoted by \(E_ n\), their subsemigroup lattices are denoted by \(\text{Sub }T_ n\) and \(\text{Sub }E_ n\). The main theorem is the following: For a finite lattice \(L\) the following conditions are equivalent: (1) \(L\) is embeddable in the subsemigroup lattice of a free semigroup; (2) \(L\) is embeddable in the subsemigroup lattice of a free commutative semigroup; (3) \(L\) is embeddable in the subsemigroup lattice of the infinite cyclic semigroup; (4) \(L\) is embeddable in the subsemigroup lattice of a free commutative nilsemigroup of index two; (5) \(L\) is embeddable in \(\text{Sub }T_ n\) for some positive integer \(n\); (6) \(L\) is embeddable in the subsemigroup lattices of a free semilattice; (7) \(L\) is embeddable in \(\text{Sub }E_ n\) for some positive integer \(n\); (8) \(L\) is embeddable in the subsemigroup lattice of a finite semilattice; (9) \(L\) is embeddable in the subsemigroup lattice of a finite nilpotent semigroup; (10) \(L\) is lower bounded.
reviewed by: Bohdan Zelinka
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 06B25
zbMATH DE Number: 164976
zbMATH Keywords: lower bounded lattices
zbMATH Keywords: free lattice
zbMATH Keywords: subsemigroup lattices
zbMATH Keywords: finite lattice
zbMATH Keywords: embeddable in the subsemigroup lattice
zbMATH Keywords: free commutative nilsemigroup of index two
zbMATH Keywords: free semilattice
zbMATH Keywords: finite nilpotent semigroup
author: Vladimir B. Repnitskiǐ
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 164976