@MISC{Eberhart_arepresentation, author = {Carl Eberhart and Wiley Williams}, title = {A REPRESENTATION OF THE FREE ELEMENTARY ORTHODOX SEMIGROUP}, year = {} }

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Abstract

Abstract. The free elementary inverse semigroup 27 has a simple representa-tion as a semigroup of transformations on the set of integers. In this note, we obtain a fairly simple representation of a pre-image of 27, the free elementary orthodox semigroup (9. Let (9(I) denote the free elementary orthodox (inverse) semigroup on the letters p and q, that is, O = (p, q Ipnq'•p ' • = p•, q•p•q' • = q • for each n> 0) (l(1) foreach,>O,p"q"p'•=p'•,q'•p"q'•=q' • I I-- P ' q (2) r each n,m>0, pnqn+,•p,•____qmp,•+mqn Z can be realized as a subsemigroup of the semigroup f(Z) of trans-formations on the integers by using the function defined by p(n) = n + 1 (n •-1) and p(-1) =-1, and its inverse q in the semigroup f(Z) defined by q(n) = n- 1 (n • 0) and q(0) = 0. In [2] the authors raised the question of whether O could be represented as a subsemigroup of f(Z). The semigroup O was studied in [2], where it was shown that each element of can be written uniquely in the form of an alternating product i2. il of powers of p and q, w*•x2..x•. with {X•_l,X•}={p,q} forq < k_<l, and i •> min(i•_l,i•+l) for 1 < k < l- 1. Upon inspection, it can be seen that words written in this canonical form have the shape of a mountain. By this we mean that the sequence of exponents il,..., it increases to a maximum value which is assumed at most twice and then decreases. Refer to the maximum exponent, say n, occuring in a mountain as the order of the mountain. Then each mountain has a west il iw where ' < i • = n, and an east slope x? x I ' where slope, x 1 "'xw, zw-1 ß ", i, = n> i,_•. Green's relations on O are then determined as follows (see