A note on infinitely distributive inverse semigroups
Pedro Resende · 2012
By an infinitely distributive inverse semigroup will be meant an inverse semigroup S such that for every subset X ⊆ S and every s ∈ S, if ∨ X exists then so does ∨ (sX), and furthermore ∨ (sX) = s ∨ X. One important aspect is that the infinite distributivity of E(S) implies that of S; that is, if the multiplication of E(S) distributes over all the joins that exist in E(S) then S is infinitely distributive. This can be seen in Proposition 20, page 28, of Lawson’s book [1]. Although the statement of the proposition mentions only joins of nonempty sets, the proof applies equally to any subset. The aim of this note is to present a proof of an analogous property for binary meets instead of multiplication; that is, we show that for any infinitely distributive inverse semigroup the existing binary meets distribute over all the joins that exist. A useful consequence of this lies in the possibility of constructing, from infinitely distributive inverse semigroups, certain quantales that are also locales (due to the stability of the existing joins both with respect to the multiplication and the binary meets), yielding a direct connection to étale groupoids via the results of [2]. The consequences of this include an algebraic construction of “groupoids of germs ” from certain inverse semigroups, such as pseudogroups, and will be developed elsewhere. Lemma. Let S be an inverse semigroup, and let x,y ∈ S be such that the meet x ∧ y exists. Then the join f = ∨ {g ∈ ↓(xx −1 yy −1) | gx = gy} exists, and we have x ∧ y = fx = fy. Proof. Consider the set Z of lower bounds of x and y, Z = {z ∈ S | z ≤ x, z ≤ y},