Errata to “ Semigroups , rings , . . . ” April 10 , 2016 Semigroups , rings , and Markov chains

I will refer to the results appearing in the paper " Semigroups, rings, and Markov chains " by the numbers under which they appear in this paper (specifically , in its version of 20 June 2000, which appears on arXiv as preprint arXiv:math/0006145v1). Errata • Page 1, §1: " explcitly " → " explicitly ". • Page 3, §1.4: Replace " q-elements " by " q elements ". (The hyphen should not be there.) • Page 6, §2.1: At the end of the first paragraph on this page (i.e., right after the sentence " Thus S has the deletion property (D) stated in Section 1.1 "), I suggest adding something like the following definitions: " When S is a LRB, the lattice L and the map supp in the above definition are determined uniquely up to isomorphism (this follows easily from the surjectivity of supp and the equivalence (3)). The lattice L is called the lattice of supports (or the support lattice) of S. The surjection supp : S → L is called the support map of S. For any s ∈ S, the element supp s of L is called the support of s. " (Of course, the purpose of these sentences is to introduce some notations that you use several times; you'll probably find a more succinct way to introduce them.) • Page 6, §2.1: The last comma in " Sections 4, 5, and 6, " probably should be a period. • Page 6, §2.2: You write that S ≥x " is a LRB in its own right, the associated lattice being the interval X, 1 in L, where X = supp x ". This is correct, but (in my opinion) not obvious enough to be left to the reader. • Page 6, §2.2: You write that S ≥x " is a LRB in its own right, the associated lattice being the interval X, 1 in L, where X = supp x ". This is correct, but (in my opinion) not obvious enough to be left to the reader. 1 1 Here is a proof, for the sake of completeness: