Universality of the lattice of transformation monoids. (English) Zbl 1286.06009
The authors solve an open problem in [M. Goldstern and M. Pinsker, Algebra Univers. 59, No. 3–4, 365–403 (2008; Zbl 1201.08003)]:
Theorem 1.1: Mon(\(\lambda\)) is universal for complete algebraic lattices with at most \(2^{\lambda}\) compact elements with respect to closed sublattices; i.e., the closed sublattices on Mon(\(\lambda\)) are, up to isomorphism, precisely the complete algebraic lattices with at most \(2^{\lambda}\) compact elements.
As a corollary, they also prove that if \(L\) is an algebraic lattice with at most \(2^{\lambda}\) compact elements, then it is even isomorphic to a closed sublattice of Mon(\(\lambda\)) via an isomorphism which preserves the smallest element.
Let \(\lambda\) be a fixed infinite set of cardinality \(\lambda\) (the set is identified with its cardinality). A subset of \(\lambda^{\lambda}\) which is closed under composition and contains the identity function is called a transformation monoid on \(\lambda\) and denoted by Mon(\(\lambda\)).
Theorem 1.1: Mon(\(\lambda\)) is universal for complete algebraic lattices with at most \(2^{\lambda}\) compact elements with respect to closed sublattices; i.e., the closed sublattices on Mon(\(\lambda\)) are, up to isomorphism, precisely the complete algebraic lattices with at most \(2^{\lambda}\) compact elements.
As a corollary, they also prove that if \(L\) is an algebraic lattice with at most \(2^{\lambda}\) compact elements, then it is even isomorphic to a closed sublattice of Mon(\(\lambda\)) via an isomorphism which preserves the smallest element.
Let \(\lambda\) be a fixed infinite set of cardinality \(\lambda\) (the set is identified with its cardinality). A subset of \(\lambda^{\lambda}\) which is closed under composition and contains the identity function is called a transformation monoid on \(\lambda\) and denoted by Mon(\(\lambda\)).
Reviewer: Michiro Kondo (Inzai)
MSC:
| 06B15 | Representation theory of lattices |
| 06B23 | Complete lattices, completions |
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
Citations:
Zbl 1201.08003References:
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