Characterizing categorically closed commutative semigroups

Absolutely closed semigroups

“…Therefore, semigroups possessing the above properties are called algebraically closed. Algebraically closed commutative semigroups were investigated by the authors in [9]. Countable algebraically closed groups and semigroups were investigated in [10].…”

“…A semigroup X is C-closed if its semilattice reflection X/ is chain-finite and every maximal ¾-trivial subsemigroup x of X is C-closed. Now we are going to recall a characterization of C-closed commutative semigroups, proved by the authors in [9]. For its formulation we need the following definitions.…”

“…The following characterization of chain-finite semilattices was proved by the authors in [7] and [9].…”

“…Theorem 1.3 implies that each subsemigroup of a C-closed commutative semigroup is C-closed. On the other hand, quotient semigroups of C-closed commutative semigroups are not necessarily C-closed (see Example 1.8 from [9]). This motivates the following definitions.…”

“…By Theorem 1.4, the semigroup Z(X) is periodic and so is the group H ⊆ Z(X). Since H is not bounded, we can apply Lemma 5.10 of [9] and find a homomorphism h : H → T whose image h[H] is infinite and hence dense in T. Consider the projection pr h : K → T, pr h : y → y(h), of K onto the h-th factor and observe that pr h [K] is a compact subgroup of T that contains the dense subgroup h[H] of T. The compactness of K ensures that pr h [K] = T and hence K contains an element of infinite order. Since K is topologically isomorphic to the closure H of H in X, we can assume that the element a ∈ H \ H ⊆ X \ X is of infinite order.…”