I recently started looking into the subject of foundation semi-groups and I'm trying to find simple (none-trivial) examples of foundations semi-groups.
I know that every locally compact group is foundation and discrete semi-groups are also foundation. Is there another examples?
Thank you
According to [1], every open subsemigroup of a locally compact group is a foundation semigroup. Moreover, every finite product of foundation semigroups and any image of a foundation semigroup under a continuous and proper homomorphisms is a foundation semigroup.
Another example can be found in [2, Example 2.1].
[1] John W. Baker, Measure algebras on semigroups. The analytical and topological theory of semigroups, 221--252, de Gruyter Exp. Math., 1, de Gruyter, Berlin, 1990.
[2] Henry A. M. Dzinotyiweyi, Gérard L. G. Sleijpen, A note on measures on foundation semigroups with weakly compact orbits. Pacific J. Math. 81 (1979), no. 1, 61--69.