Maximality and totality of stable functions in the category of stable bifinite domains

This paper studies maximality and totality of stable functions in the category of stable bifinite domains. We present three main results as follows, (1) every maximum-preserving function is a maximal element in the stable function spaces; (2) a maximal stable function f : D → E is maximum-preserving if D is maximum-separable and E is completely separable; and (3) a stable bifinite domain D is maximum-separable if and only if for any locally distributive stable bifinite domain E, each maximal stable function f : D → E is maximum-preserving.