Real subset sums and posets with an involution

Author:

Bisi Cinzia1,Chiaselotti Giampiero2,Gentile Tommaso2

Affiliation:

1. Dipartimento di Matematica e Informatica, Universitá di Ferrara, Via Machiavelli 30, 44121, Ferrara, Italy

2. Dipartimento di Matematica, Universitá della Calabria, Via Pietro Bucci, Cubo 30B, 87036 Arcavacata di Rende (CS), Italy

Abstract

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

Publisher

World Scientific Pub Co Pte Ltd

Subject

General Mathematics

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