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regular semigroup

regular, $\pi$-regular, eventually regular, strongly $\pi$-regular, group-bound, inverse semigroup, Clifford semigroup, orthodox semigroup, completely regular, epigroup, regular element, inverse, relative inverse
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Mathematics Subject Classification

20M17 no label found20M18 no label found


seems like a completely regular semigroup is a group

HkBst writes:

> seems like a completely regular semigroup is a group


Let G_1 and G_2 be two (distinct) groups. Let S be the set formed as the disjoint union of G_1 and G_2. Add a new element 0.

Leave the product of two elements in G_1 as before, and the same with G_2. Define the product of an element in G_1 with an element in G_2 as 0, and vice versa. Let 0 act as a zero element. (This is called, in semigroup theory, the 0-disjoint union)

Now, {0} is a subgroup of S, as are G_1 and G_2. Every element is in one of these, so S is completely regular. However, S is clearly not a group.

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