N ov 2 00 7 POSET REPRESENTATIONS OF DISTRIBUTIVE

3 N ov 2 00 7 POSET REPRESENTATIONS OF DISTRIBUTIVE SEMILATTICES

We prove that for every distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (P1) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (P2) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) ≤ a ∨ b, then there are a pos...

N ov 2 00 7 Representations of quivers , their generalizations and invariants

This paper is a survey on invariants of representations of quivers and their generalizations. We present the description of generating systems for invariants and relations between generators.

ar X iv : m at h . C O / 0 41 16 10 v 1 2 7 N ov 2 00 4 CHAIN POLYNOMIALS OF DISTRIBUTIVE LATTICES ARE 75 %

It is shown that the numbers ci of chains of length i in the proper part L \ {0, 1} of a distributive lattice L of length l + 2 satisfy the inequalities c0 < . . . < c⌊l/2⌋ and c⌊3l/4⌋ > . . . > cl. This proves 75 % of the inequalities implied by the Neggers unimodality conjecture.

. R A ] 4 J an 2 00 6 POSET REPRESENTATIONS OF DISTRIBUTIVE SEMILATTICES

We prove that for any distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (i) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (ii) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) = a ∨ b, then there are a positi...

1 5 N ov 2 00 2 Representations of SL 2 and the distribution of points in