N5 as a Subalgebra of a Principally Ordered Naturally Ordered Regular Semigroup with a Zero Element

We start with a basic and technical result: If S is a principally ordered regular semigroup that has a zero element, 0, then 0* is the biggest element (in fact, idempotent) of S. We obtain necessary and sufficient conditions for a principally ordered naturally ordered regular semigroup, S, with a zero element, to have as a subalgebra an ordered and algebraic isomorphic copy of N5, the smallest naturally ordered non-orthodox regular semigroup with a biggest idempotent.

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