Naturally Totally Ordered Commutative Semigroups

AnL1algebra for linearly quasi-ordered compact semigroups

“…Set T = iS 0 /^/ r and let ^ be the natural map. Then T is a naturally totally ordered semigroup in the sense of Clifford [1] and is a compact topological semigroup with 0 and 1 and no other idempotents in the order topology (which agrees with the original topology). If 0 ^ α, 6, c ^ 1 in T and ac -be then (since a ^ b or b ^ α, we assume 6 ^ α) there is a ώ e Γ such that b -ad.…”

“…Let E denote the idempotent elements of S and for each e e E, let H(e) be the maximal subgroup of S containing e. Since S is commutative with identity, Se is the principal ideal generated by e. If e, fe E and Se c Sf, then we write e ^ / and note that e ^ / if and only if ef = e. It is also clear that E, with ^, is a naturally totally ordered set in the sense of Clifford [1]. y.…”

“…Let S = [0,2] where [0,1] is a usual unit interval, [1,2] is a usual unit interval and each element of [1,2] …”

“…Let S = [0,3] where [0,1] is a usual unit interval and [2,3] is a usual unit interval but [1,2] is a continuum of idempotent elements and each interval acts as identity for the ones below…”

ℋ-commutative semigroups in which each homomorphism is uniquely determined by its kernel

“…Indeed, it is easy to see that g could be taken to be any element of the group, and (1) would still be true. On the other hand, (1) does not hold for arbitary semigroups. For example, we shall see (as a consequence of Lemma 1) that (1) does not hold for the positive integers, if we take either multiplication or addition as the semigroup operation.…”