The inheritance of symmetry conditions in commutative semigroup cohomology

Cochains in commutative semigroup cohomology satisfy symmetry conditions that have only been ascertained in dimensions n≤4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \le 4$$\end{document}. This article studies how symmetry conditions are passed from one dimension to the next, and proposes two conjectures about the appropriate conditions for dimensions 5 and 6.