The terms posemigroup, pogroup and pomonoid are sometimes used, where "po" is an abbreviation for "partially ordered".
Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=".
A morphism or homomorphism of posemigroups is a semigroup homomorphism that preserves the order (equivalently, that is monotonically increasing).
A pomonoid (M, •, 1, ≤) can be considered as a monoidal category that is both skeletal and thin, with an object of for each element of M, a unique morphism from m to n if and only if m ≤ n, the tensor product being given by •, and the unit by 1.
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