Given a finite partially ordered set \((A,\leq)\), an algebra \(\mathcal A\) defined on this set is said to be order-primal if every finitary function on \(A\) which preserves the given partial order is a term function of \(\mathcal A\). The present paper considers the case when the set \(A\) is connected under \(\leq\), which means that any two elements \(a,b\) of \(A\) can be connected by a sequence of elements starting with \(a\) and ending in \(b\) such that any two neighbouring elements of the sequence are comparable. In this case it is easy to deduce from known results that the algebra \(\mathcal A\) is simple, has no proper subalgebra and no non-identical automorphism, and every constant function is a term function of \(\mathcal A\) (the last of these properties is meant by saying that \(\mathcal A\) is constantive); furthermore, \(\mathcal A\) must have at least two fundamental operations which are not both unary. After this, the authors investigate the question when such an algebra generates a minimal variety or quasivariety, and also obtain new primality criteria for algebras whose set of term functions contains all functions which preserve a connected unbounded order on the base set of the algebra.