We study countably compact and absolutely H-closed primitive topological inverse semigroups. We describe the structure of compact and countably compact primitive topological inverse semigroups and show that any countably compact primitive topological inverse semigroup embeds into a compact primitive topological inverse semigroup.In this paper all spaces are Hausdorff. A semigroup is a nonempty set with a binary associative operation. A semigroup S is called inverse if for any x ∈ S there exists a unique y ∈ S such that x · y · x x and y · x · y y. Such an element y in S is called inverse to x and denoted by x −1 . The map defined on an inverse semigroup S which maps to any element x of S its inverse x −1 is called the inversion. A topological semigroup is a Hausdorff topological space with a jointly continuous semigroup operation. A topological semigroup which is an inverse semigroup is called an inverse topological semigroup. A topological inverse semigroup is an inverse topological semigroup with continuous inversion. A topological group is a topological space with a continuous group operation and an inversion. We observe that the inversion on a topological inverse semigroup is a homeomorphism see 1, Proposition II.1 . A Hausdorff topology τ on a inverse semigroup S is called inverse semigroup if S, τ is a topological inverse semigroup.Further we shall follow the terminology of 2-8 . If S is a semigroup, then by E S we denote the band the subset of idempotents of S, and by S 1 S 0 we denote the semigroup S with the adjoined unit zero see 7, page 2 . Also if a semigroup S has zero 0 S , then for any A ⊆ S we denote A * A \ {0 S }. If Y is a subspace of a topological space X and A ⊆ Y , then

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Students connect operations in linear algebra with geometric representations.

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