Absolutely flat semigroups

Left orders in regular ℋ-Semigroups II

“…But it follows from the fact that 0 is an 2T-class and SI%C* is categorical at 0 that 5 is categorical at 0: thus r(ab) = r(b). Let x, y eS 1 By Theorem 2.4, 5 is a stratified left order in a regular ^-semigroup Q. But it is easy to deduce from Proposition 2.5 and Corollary 2.7 that Q is a Brandt semigroup.…”

Recreational mathematics

“…Here the five points (29,1), (23,2), (17,3), (11,4), (5,5) are at distances 1, 2, 3, 4, 5 from the hypotenuse. Another example: with paramters (5,2) we have the triangle (21,20,29).…”

On left absolutely flat bands

“…For any semigroup S, S-Ens (respectively, Ens-5) will denote the class of all left (right) S-sets and S1 will represent the monoid obtained by adjoining a new identity element 1 to S. Let A E Ens-S, a, a' E A, B E 5-Ens, and b,b' E B. Following Lemma 1.2 of [1] it is easy to show that a Cg> 6 = a' ® b' in A 0 B if and only if there exist 01,..., an E A, b2, ■ ■ ■ ,bn E B, si,... ,sn, ti,... ,tn E S1 such that a = aiSi aiti = <z2s2 sib = tib2…”

“…From this description we see that a left S-set B is flat if and only if, for every right 5-set A, and every a, a' E A, b,b' E B such that there exists a scheme over A and B joining (a, b) to (a',br), there exists a scheme (of possibly different length) over aS1 Ua'S1 and B joining (a, b) to (a',b'). (See Lemma 2.2 of [1].) In the sequel S = IJ-fS-y | 7 E Y} will denote a right regular band with its associated greatest semilattice decomposition (r is a semilattice, each S-, is a right zero semigroup, and S-ySg Ç S1s for each 7, S E Y).…”

“…Then S is a semilattice of rectangular bands, and if S is left absolutely flat then each rectangular component must be a right zero band ( [3], or an easy adaptation of [1,Theorem 4.3]). Therefore every left absolutely flat band is right regular (the variety of right regular bands is defined by the equation xyx = yx).…”