Shifting operations and completely $t$-spread lexsegment ideals

TL;DR

This paper introduces and studies the properties of arbitrary t-spread lexsegment ideals, generalizing classical lexsegment ideals, and characterizes those with linear resolutions.

Contribution

It defines arbitrary t-spread lexsegment ideals, characterizes completely t-spread lexsegment ideals, and classifies those with linear resolutions.

Findings

Characterization of all completely t-spread lexsegment ideals.

Classification of completely t-spread lexsegment ideals with linear resolution.

Generalization of lexsegment ideals to t-spread context.

Abstract

In this paper we introduce the concepts of arbitrary $t$-spread lexsegments and of arbitrary $t$-spread lexsegment ideals with $t$ a positive integer. These concepts are a natural generalization of arbitrary lexsegments and arbitrary lexsegment ideals. An ideal generated by an arbitrary $t$-spread lexsegment is called completely $t$-spread lexsegment if it is equal to the intersection of an initial $t$-spread lexsegment ideal and of a final $t$-spread lexsegment ideal. We study the class of arbitrary $t$-spread lexsegment ideals. In particular, we characterize all completely $t$-spread lexsegment ideals. Moreover, we classify all completely $t$-spread lexsegment ideals with a linear resolution.