Sublattices of Lattices of Order-Convex Sets, Iii: The Case of Totally Ordered Sets

Semenova, Marina N.; Wehrung, Friedrich

doi:10.1142/s021819670400175x

Cited by 17 publications

(20 citation statements)

References 9 publications

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“…Our proofs of the main results are underpinned by the technique of colored trees, which turned out to be very useful in proving different embeddability results in [4,[8][9][10][11]. The definition of a colored tree first appeared in an explicit form in [11].…”

Section: Preliminariesmentioning

confidence: 99%

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Lattices embeddable in subsemigroup lattices. I. Semilattices

Semenova

2006

Algebr Logic

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UDC 512.56Keywords: lattice, subsemilattice lattice, lower bounded lattice, partial order.V. B. Repnitskii showed that any lattice embeds in some subsemilattice lattice. In his proof, use was made of a result by D. Bredikhin and B. Schein, stating that any lattice embeds in the suborder lattice of a suitable partial order. We present a direct proof of Repnitskii's result, which is independent of Bredikhin-Schein's, giving the answer to a question posed by L. N. Shevrin and A. J. Ovsyannikov. We also show that a finite lattice is lower bounded iff it is isomorphic to the lattice of subsemilattices of a finite semilattice that are closed under a distributive quasiorder. PRELIMINARIESFor any semilattice P, ∧ and any set X ⊆ P , X denotes the subsemilattice of P generated by X. A direct check shows that X = {s 0 ∧ . . . ∧ s n | n < ω and s i ∈ P for all i n}.For any semilattice P, ∧ , by Sub P we denote the subsemilattice lattice of P, ∧ . Sub P forms a complete algebraic lattice under set-theoretic inclusion in whichFollowing [1], we say that a class U of algebraic structures (of a fixed signature) is lattice-universal if any lattice embeds in the lattice Sub A of substructures of some structure A ∈ U. Several examples of lattice-universal classes of structures are known to date. Among these are the class of all groups [2], the class of partially ordered sets [3] (see also [4]), and various classes of semigroups [5]. The results on lattice-universal classes of semigroups in [5] are collected in THEOREM 1.1 [1, Thm. 27.2]. The following classes of semigroups are lattice-universal: (i) the class of all commutative nil-semigroups of index 2; *

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Section: Preliminariesmentioning

confidence: 99%

“…The definition of a colored tree first appeared in an explicit form in [11]. Until then, however, it had been used implicitly in a series of papers dealing in lattices of convex subsets [8][9][10]; see also [12].…”

Section: Preliminariesmentioning

confidence: 99%

Lattices embeddable in subsemigroup lattices. I. Semilattices

Semenova

2006

Algebr Logic

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“…Say that a lattice is bi-algebraic, if it is both algebraic and dually algebraic. While investigating lattices of convex subsets, the authors of [21,22] came across the following problem, which is stated as Problem 5 in [21].…”

Section: Introductionmentioning

confidence: 99%

Sublattices of complete lattices with continuity conditions

Wehrung

2005

Algebra univers.

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Abstract. Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice Id S of S is both algebraic and dually algebraic. Furthermore, if there are no infinite D-sequences in J(S), then Id S can be embedded into a direct product of finite lower bounded lattices. We also find a system of infinitary identities that characterize sublattices of complete, lower continuous, and join-semidistributive lattices. These conditions are satisfied by any (not necessarily finitely generated) lower bounded lattice and by any locally finite, join-semidistributive lattice. Furthermore, they imply M. Erné's dual staircase distributivity.On the other hand, we prove that the subspace lattice of any infinitedimensional vector space cannot be embedded into any ℵ 0 -complete, ℵ 0 -upper continuous, and ℵ 0 -lower continuous lattice. A similar result holds for the lattice of all order-convex subsets of any infinite chain.

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“…A proof with which we are coming up here employs the technique of colored trees, which has turned out to be very useful for proving different embedding results (see [4,[8][9][10][11]). The definition of a colored tree appeared in its explicit form in [11].…”

Section: Preliminariesmentioning

confidence: 99%

Lattices embeddable in subsemigroup lattices. II. Cancellative semigroups

Semenova

2006

Algebr Logic

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Repnitskii proved that any lattice embeds in a subsemigroup lattice of some commutative, cancellative, idempotent free semigroup with unique roots. In that proof, use is made of a result by Bredikhin and Schein stating that any lattice embeds in a suborder lattice of suitable partial order. Here, we present a direct proof of Repnitskii's result which is independent of Bredikhin-Schein's, thus giving the answer to the question posed by Shevrin and Ovsyannikov. PRELIMINARIESFor an algebraic structure A, Sub(A) denotes the set of all substructures of A. Ordered by inclusion, Sub(A) forms a complete algebraic lattice, wheresubstructures in A. (Here, for any X ⊆ A, X denotes the substructure of A generated by X.) Following [1], we say that a class U of algebraic structures (of a fixed signature) is lattice universal if any lattice embeds in Sub(A), for some A ∈ U. At present we know of several examples of lattice-universal classes of structures. Among these are the class of all groups [2], the class of partially ordered sets [3; see also 4], and various classes of semigroups [5]. The results on lattice-universal classes of semigroups are summarized in THEOREM 1.1 [1, Thm. 27.2]. The property of being lattice universal is shared by the following classes of semigroups: (i) the class of all commutative nil semigroups of index 2; (ii) the class of all semilattices;(iii) the class of commutative, cancellative, idempotent free semigroups with unique roots. The original proof of this result in [5] made essential use of the above-mentioned result that any lattice embeds in a suborder lattice of suitable partial order. Namely, in [5], it was proved that any suborder *

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Sublattices of Lattices of Order-Convex Sets, Iii: The Case of Totally Ordered Sets

Cited by 17 publications

References 9 publications

Lattices embeddable in subsemigroup lattices. I. Semilattices

Lattices embeddable in subsemigroup lattices. I. Semilattices

Sublattices of complete lattices with continuity conditions

Lattices embeddable in subsemigroup lattices. II. Cancellative semigroups

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