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THE FREE MONOGENIC INVERSE SEMIGROUP and the BICYCLIC MULTIPLICATION 1. Introduction

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THE FREE MONOGENIC INVERSE SEMIGROUP and the BICYCLIC MULTIPLICATION 1. Introduction Ann. Sci. Math. Québec 36, No 1, (2012), 235–243 THE FREE MONOGENIC INVERSE SEMIGROUP AND THE BICYCLIC MULTIPLICATION EMIL DANIEL SCHWAB RÉSUMÉ. Dans la théorie des semi-groupes inverses, une attention spéciale est donnée aux semi-groupes inverses libres, et plus particulièrement au semi-groupe in- verse libre monogénique. Celui-ci admet un nombre intéressant de copies isomorphes et Petrich a présenté la construction de cinq d’entre elles. Dans cet article, nous pré- sentons la construction d’une nouvelle copie isomorphe du semi-groupe inverse libre monogénique dont le produit implique uniquement la multiplication bicyclique. La construction est élémentaire et fait appel à la propriété universelle d’un objet libre. Ceci permet à l’article d’être indépendant des autres constructions apparaissant dans la littérature en lien avec la théorie des semi-groupes inverses libres. ABSTRACT. In the theory of inverse semigroups, a special attention is given to free inverse semigroups, in particular to the free monogenic inverse semigroup. The free monogenic inverse semigroup admits a number of interesting isomorphic copies. Petrich has presented the constructions of five isomorphic copies of the free monogenic inverse semigroup. In this paper we give a new isomorphic copy of the free monogenic inverse semigroup with a product involving only the bicyclic multiplication. Using an elementary way based on the universal property of a free object, the paper is organized so as to be self-contained; it is not dependent on other constructions from the theory of free inverse semigroups. 1. Introduction An inverse semigroup S is a semigroup in which every element s in S has a unique generalized inverse x in S, in the sense that s = sxs and x = xsx. In inverse semi- groups we denote the unique generalized inverse x of an element s by s−1. A semigroup S is inverse if and only if each element has a generalized inverse and its idempotents commute. This is equivalent to the condition that each L -class and each R-class of S contains exactly one idempotent. The relations L and R (and D = L ◦ R = R ◦ L and H = L \ R) are the Green’s relations which can be defined on any semigroup in terms of principal ideals, but assume simple forms on inverse semigroups: sL t () s−1s = t−1t and sRt () ss−1 = tt−1: An inverse semigroup with only one idempotent is a group. Inverse semigroups were introduced independently by Wagner (1952) and Preston (1954). Wagner called inverse semigroups “generalized groups”. As the group theory, Reçu le 23 mai 2011 et, sous forme définitive, le 20 mars 2012. 236 THE FREE MONOGENIC INVERSE SEMIGROUP AND THE BICYCLIC MULTIPLICATION the inverse semigroup theory has found many applications (in differential geometry, model theory, combinatorial group theory, linear logic, formal languages etc.). The Wagner-Preston representation theorem sets out that every inverse semigroup can be faithfully represented by partial bijections. This is an analogue to Cayley’s theorem in group theory. Semigroups of partial bijections closed under inversion are examples of inverse semigroups. Another important example, further used, is the bicyclic semi- group. The bicyclic semigroup B is the bisimple (it consists of a single D-class) inverse monoid of all pairs (a; b) of non-negative integers with the multiplication (called the bi- cyclic multiplication) given by: (a; b) ◦ (a0; b0) = (a − b + maxfb; a0g; maxfb; a0g − a0 + b0): The identity element of B is 1 = (0; 0). Note that (a; b) = sa ◦ tb and t ◦ s = 1, where s = (1; 0) and t = (0; 1). The set E(B) = f(a; a) j a 2 Ng is the set of idempotents of B, and we have (a; b)−1 = (b; a). It is straightforward to check that (a; b)L (a0; b0) () b = b0 and (a; b)R(a0; b0) () a = a0: It follows that any two elements of B are D-related and H is the identity relation. Such an inverse semigroup is bisimple and aperiodic. In many types of algebraic structure the free object is one of the basic concepts. Free semigroup, free monoid, free group, free ring, free module, free lattice, free Boolean algebra, etc. are important objects in abstract algebra and related mathematics theory. The universal property of a free object (if such an object exists) characterizes uniquely (up to an isomorphism) this object. Given a non-empty set X, a pair (A; f), that is an object A of an algebraic structure together with a map f : X ! A, is the free object on X if for any object B of the same algebraic structure and any func- tion g : X ! B there exists a unique morphism h : A ! B such that the diagram commutes (see Figure 1). If X is a singleton then the free object on X is called a free FIGURE 1. The commuting diagram monogenic object. The free monogenic monoid is the additive monoid of non-negative integers N and the free monogenic group is the additive group of integers Z. It is somewhat surprising that the free monogenic inverse semigroup is not simple as is the free monogenic group or the free monogenic monoid. The first explicit construction of the free monogenic inverse semigroup was given by Gluskin [2] (see Petrich’s C5 [3, p. 400]): the set { } 3 C5 = (p; q; r) 2 Z p ≥ 0; r ≥ 0; p + q ≥ 0; q + r ≥ 0 and p + q + r > 0 together with the product (p; q; r) · (p0; q0; r0) = (maxfp; p0 − qg; q + q0; maxfr0; r − q0g): E. D. Schwab 237 Since the free monogenic inverse semigroup is isomorphic to a certain subsemigroup of the direct product of two copies of the bicyclic semigroup (see Scheiblich’s represen- tation, [5]), the bicyclic multiplication is naturally associated with the free monogenic inverse semigroup. It is known that the free monogenic inverse semigroup admits a va- riety of peculiar isomorphic copies. Some useful references for such isomorphic copies are Petrich [3, Chapter IX] and Reilly’s survey article [4]. 2. A semigroup of triples involving bicyclic multiplication In this section we will construct a new isomorphic copy of the free monogenic inverse semigroup considering the following set { } I = (a0; a; m) 2 N3 a0; a ≤ m and m =6 0 ; equipped with the product (a0; a; m) · (b0; b; n) = ((a0; a) ◦ (b0; b); maxf(m; a) ◦ (b0; n)g); where ◦ is the bicyclic multiplication of pairs of non-negative integers and where maxf(x; y)g means maxfx; yg. It is straightforward to check that the product is well- defined. In what follows we will show that the product is associative. For that, we use the following notation: ( [(a0; a; m) · (b0; b; n)] · (c0; c; p) = (x0; x; q) · (c0; c; p) = (y0; y; s); (a0; a; m) · [(b0; b; n) · (c0; c; p)] = (a0; a; m) · (z0; z; v) = (u0; u; t); and we shall prove that y0 = u0; y = u and s = t. Since (y0; y) = (x0; x) ◦ (c0; c) = [(a0; a) ◦ (b0; b)] ◦ (c0; c) = (a0; a) ◦ [(b0; b) ◦ (c0; c)] = (a0; a) ◦ (z0; z) = (u0; u); it follows that y0 = u0 and y = u. Hence, x0 − x + maxfx; c0g = a0 − a + maxfa; z0g (denoted by α) and maxfx; c0g − c0 + c = maxfa; z0g − z0 + z (denoted by β): We now prove that s = t. We have s = maxf(q; x) ◦ (c0; p)g = maxfq − x + maxfx; c0g; maxfx; c0g − c0 + pg = maxfq − x0 + x0 − x + maxfx; c0g; maxfx; c0g − c0 + c + p − cg = maxfq − x0 + α; β + p − cg 238 THE FREE MONOGENIC INVERSE SEMIGROUP AND THE BICYCLIC MULTIPLICATION and t = maxf(m; a) ◦ (z0; v)g = maxfm − a + maxfa; z0g; maxfa; z0g − z0 + vg = maxfm − a0 + a0 − a + maxfa; z0g; maxfa; z0g − z0 + z + v − zg = maxfm − a0 + α; β + v − zg: After some calculations, we have ( m − a0 if m − a ≥ n − b0; q − x0 = n − a0 + a − b0 if m − a ≤ n − b0 and ( n − b + c0 − c if n − b ≥ p − c0; v − z = p − c if n − b ≤ p − c0: Case 1: Assume that m − a ≥ n − b0 and n − b ≤ p − c0. Then, s = maxfq − x0 + α; β + p − cg = maxfm − a0 + α; β + p − cg = maxfm − a0 + α; β + v − zg = t: Case 2: Assume that m − a ≤ n − b0 and n − b ≥ p − c0. After some calculations we obtain n − a0 + a − b0 + α = β + n − b + c0 − c ≥ m − a0 + α ; β + p − c: It follows that s = maxfq − x0 + α; β + p − cg = maxfn − a0 + a − b0 + α; β + p − cg = n − a0 + a − b0 + α = β + n − b + c0 − c = maxfm − a0 + α; β + n − b + c0 − cg = maxfm − a0 + α; β + v − zg = t: Case 3: Assume that m − a ≥ n − b0 and n − b ≥ p − c0. The following inequalities hold: m − a0 + α ≥ β + p − c and m − a0 + α ≥ β + v − z: E. D. Schwab 239 It follows that s = maxfq − x0 + α; β + p − cg = maxfm − a0 + α; β + p − cg = m − a0 + α = maxfm − a0 + α; β + v − zg = t: Case 4: Assume that m − a ≤ n − b0 and n − b ≤ p − c0. Then, n − a0 + a − b0 + α ≤ β + p − c and m − a0 + α ≤ β + p − c; and therefore s = maxfq − x0 + α; β + p − cg = maxfn − a0 + a − b0 + α; β + p − cg = β + p − c = maxfm − a0 + α; β + p − cg = maxfm − a0 + α; β + v − zg = t: Thus we have proved associativity. The set { } E(I) = (a; a; m) 2 N3 a ≤ m and m =6 0 is the set of idempotents of I.
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