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Totally compatible structures on the radical of an incidence algebra

Mykola Khrypchenko

TL;DR

The paper classifies totally compatible bilinear structures on the Jacobson radical of the incidence algebra $J(I(X,K))$ for finite posets $X$. It develops a framework based on centroid-valued mutations and annihilator-valued structures, yielding an explicit description of proper tot-comp structures and a complete decomposition of all tot-comp structures via $*_{\\mathcal{C}}$ indexed by $X^3_{<}/\\approx$ classes. A sufficient condition guarantees all tot-comp are proper, while examples show non-proper and annihilator-valued cases, including a sharp result when $l(X)\\le 2$. An open problem remains to characterize posets for which every tot-comp structure on $J(I(X,K))$ is proper, highlighting rich combinatorial-algebraic interplay between the poset geometry and compatible algebraic structures.

Abstract

We describe totally compatible structures on the Jacobson radical of the incidence algebra of a finite poset over a field. We show that such structures are in general non-proper.

Totally compatible structures on the radical of an incidence algebra

TL;DR

The paper classifies totally compatible bilinear structures on the Jacobson radical of the incidence algebra for finite posets . It develops a framework based on centroid-valued mutations and annihilator-valued structures, yielding an explicit description of proper tot-comp structures and a complete decomposition of all tot-comp structures via indexed by classes. A sufficient condition guarantees all tot-comp are proper, while examples show non-proper and annihilator-valued cases, including a sharp result when . An open problem remains to characterize posets for which every tot-comp structure on is proper, highlighting rich combinatorial-algebraic interplay between the poset geometry and compatible algebraic structures.

Abstract

We describe totally compatible structures on the Jacobson radical of the incidence algebra of a finite poset over a field. We show that such structures are in general non-proper.
Paper Structure (14 sections, 24 theorems, 72 equations)

This paper contains 14 sections, 24 theorems, 72 equations.

Table of Contents

  1. Definitions and preliminaries
  2. Matching compatibilities and interchangeability
  3. Isomorphic and antiisomorphic structures
  4. Posets
  5. Incidence algebras
  6. Some general properties of totally compatible structures
  7. Proper totally compatible structures
  8. Proper totally compatible structures on $(J(I(X,K)),\cdot)$
  9. The centroid of $J(I(X,K))$
  10. The description of proper totally compatible structures
  11. Totally compatible structures on $(J(I(X,K)),\cdot)$
  12. A class of totally compatible structures
  13. The description of totally compatible structures
  14. Open problem

Key Result

Lemma 2.1

Let $\{\cdot_i\}_{i\in I}$ be a family of bilinear products on a same vector space $V$. If $\{\cdot_i\}_{i\in I}$ are pairwise $\sigma$-matching (resp., interchangeable or totally compatible), then any two finite linear combinations of $\{\cdot_i\}_{i\in I}$ are $\sigma$-matching (resp., interchange

Theorems & Definitions (65)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Definition 3.4
  • Remark 3.5
  • Remark 3.6
  • ...and 55 more