Products of idempotent endomorphisms of an order-independence algebra of finite rank (Q1375881)

Let \((A,P)\) be a pair where \(A\) is an algebra and \(P\subseteq A\) is a partially ordered set with partial order \(\leq\). The pair \((A,P)\) is said to be an order-independence algebra if it satisfies a number of additional conditions. Denote by \(\text{End}_\leq(A)\) the monoid of all order preserving endomorphisms of \(A\) and let \(\Aut_\leq(A)\) denote the group of all order preserving automorphisms of \(A\). In the main result, the author proves that \(\text{End}_\leq(A)\setminus\Aut_\leq(A)\) is generated by its idempotents. Examples of order-independence algebras include finite Boolean algebras, chains, and finite direct powers of division rings considered as algebras over those rings. Because of this, the author's result generalizes results of previous authors concerning the endomorphism semigroups of finite chains and finite Boolean algebras.