Let \((A,P)\) be a pair where \(A\) is an algebra and \(P\subseteq A\) is a poset with order relation \(\leq \). A set \(X\subseteq A\) is said to be independent if, for all \(x\in X\), we have \(x\notin \text{Sg}(X\setminus \{x\})\). The definition of order-independence algebra is introduced by abstracting some properties of chains in finite Boolean algebras. Finite Boolean algebras, chains and finite direct powers of division rings are natural examples of order-independence algebras. A chain \(C\) in \(A\) is said to be an ordered basis for \(A\) if \(C\) is independent and \(\text{Sg} (C) = A\). The notion of the rank of an order-independence algebra can be defined as the cardinality of an ordered basis and it is proved that subalgebras of order-independence algebras of a finite rank are also order-independence algebras. Also the monoid \(\text{End}_{ \leq } (A)\) of all order-preserving endomorphisms of \(A\) of a finite rank is described.