arXiv subjects

Marina V. Semenova

Publications and source records attributed to Marina V. Semenova.

4 recordsLinked to original sources

arXiv2005

Sublattices of lattices of order-convex sets, III. The case of totally ordered sets

For a partially ordered set P, let Co(P) denote the lattice of all order-convex subsets of P. For a positive integer n, we denote by SUB(LO) (resp., SUB(n)) the class of all lattices that can be embedded into a product of lattices of convex subsets of chains (resp., chains with at most n elements). We prove the following results: (1) Both classes SUB(LO) and SUB(n), for any positive integer n, are locally ﬁnite, ﬁnitely based varieties of lattices, and we ﬁnd ﬁnite equational bases of these varieties. (2) The variety SUB(LO) is the quasivariety join of all the varieties SUB(n), for 1 ≤ n < ω, and it has only countably many subvarieties. We classify these varieties, together with all the ﬁnite subdirectly irreducible members of SUB(LO). (3) Every ﬁnite subdirectly irreducible member of SUB(LO) is projective within SUB(LO), and every subquasivariety of SUB(LO) is a variety.

math.GM↗

arXiv2005

Sublattices of lattices of order-convex sets, II. Posets of finite length

For a positive integer n, we denote by SUB (resp., SUBn) the class of all lattices that can be embedded into the lattice Co(P) of all order-convex subsets of a partially ordered set P (resp., P of length at most n). We prove the following results: (1) SUBn is a finitely based variety, for any n ≥ 1. (2) SUB2 is locally finite. (3) A finite atomistic lattice L without D-cycles belongs to SUB iff it belongs to SUB2; this result does not extend to the nonatomistic case. (4) SUBn is not locally finite for n ≥ 3.

math.GM↗

arXiv2005

Sublattices of lattices of order-convex sets, I. The main representation theorem

For a partially ordered set P, we denote by Co(P) the lattice of order-convex subsets of P. We find three new lattice identities, (S), (U), and (B), such that the following result holds. Theorem. Let L be a lattice. Then L embeds into some lattice of the form Co(P) iff L satisfies (S), (U), and (B). Furthermore, if L has an embedding into some Co(P), then it has such an embedding that preserves the existing bounds. If L is finite, then one can take P finite, of cardinality at most $2n^2-5n+4$, where n is the number of join-irreducible elements of L. On the other hand, the partially ordered set P can be chosen in such a way that there are no infinite bounded chains in P and the undirected graph of the predecessor relation of P is a tree.

math.GM↗

arXiv2005

Sublattices of lattices of convex subsets of vector spaces

For a left vector space V over a totally ordered division ring F, let Co(V) denote the lattice of convex subsets of V. We prove that every lattice L can be embedded into Co(V) for some left F-vector space V. Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional, and L embeds into a finite lower bounded lattice of the form $Co(V,Z)=\{X\cap Z | X\in Co(V)\}$, for some finite subset $Z$ of $V$. In particular, we obtain a new universal class for finite lower bounded lattices.

math.GM↗