Ann. Sci. Math. Québec 36, No 1, (2012), 235–243

THE FREE MONOGENIC INVERSE 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 , a special attention is given to free inverse semigroups, in particular to the free monogenic . 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 THEFREEMONOGENICINVERSESEMIGROUPANDTHEBICYCLICMULTIPLICATION 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 of all pairs (a, b) of non-negative integers with the multiplication (called the bi- cyclic multiplication) given by: (a, b) ◦ (a′, b′) = (a − b + max{b, a′}, max{b, a′} − a′ + b′). 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) = {(a, a) | a ∈ N} is the set of idempotents of B, and we have (a, b)−1 = (b, a). It is straightforward to check that (a, b)L (a′, b′) ⇐⇒ b = b′ and (a, b)R(a′, b′) ⇐⇒ a = a′. 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 group, free ring, free module, free , 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) ∈ Z p ≥ 0, r ≥ 0, p + q ≥ 0, q + r ≥ 0 and p + q + r > 0 together with the product (p, q, r) · (p′, q′, r′) = (max{p, p′ − q}, q + q′, max{r′, r − q′}). 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 = (a′, a, m) ∈ N3 a′, a ≤ m and m ≠ 0 , equipped with the product (a′, a, m) · (b′, b, n) = ((a′, a) ◦ (b′, b), max{(m, a) ◦ (b′, n)}), where ◦ is the bicyclic multiplication of pairs of non-negative integers and where max{(x, y)} means max{x, y}. 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: { [(a′, a, m) · (b′, b, n)] · (c′, c, p) = (x′, x, q) · (c′, c, p) = (y′, y, s), (a′, a, m) · [(b′, b, n) · (c′, c, p)] = (a′, a, m) · (z′, z, v) = (u′, u, t), and we shall prove that y′ = u′, y = u and s = t. Since (y′, y) = (x′, x) ◦ (c′, c) = [(a′, a) ◦ (b′, b)] ◦ (c′, c)

= (a′, a) ◦ [(b′, b) ◦ (c′, c)] = (a′, a) ◦ (z′, z)

= (u′, u), it follows that y′ = u′ and y = u. Hence, x′ − x + max{x, c′} = a′ − a + max{a, z′} (denoted by α) and max{x, c′} − c′ + c = max{a, z′} − z′ + z (denoted by β). We now prove that s = t. We have s = max{(q, x) ◦ (c′, p)}

= max{q − x + max{x, c′}, max{x, c′} − c′ + p}

= max{q − x′ + x′ − x + max{x, c′}, max{x, c′} − c′ + c + p − c}

= max{q − x′ + α, β + p − c} 238 THEFREEMONOGENICINVERSESEMIGROUPANDTHEBICYCLICMULTIPLICATION and t = max{(m, a) ◦ (z′, v)}

= max{m − a + max{a, z′}, max{a, z′} − z′ + v}

= max{m − a′ + a′ − a + max{a, z′}, max{a, z′} − z′ + z + v − z}

= max{m − a′ + α, β + v − z}. After some calculations, we have { m − a′ if m − a ≥ n − b′, q − x′ = n − a′ + a − b′ if m − a ≤ n − b′ and { n − b + c′ − c if n − b ≥ p − c′, v − z = p − c if n − b ≤ p − c′. Case 1: Assume that m − a ≥ n − b′ and n − b ≤ p − c′. Then, s = max{q − x′ + α, β + p − c}

= max{m − a′ + α, β + p − c}

= max{m − a′ + α, β + v − z} = t.

Case 2: Assume that m − a ≤ n − b′ and n − b ≥ p − c′. After some calculations we obtain n − a′ + a − b′ + α = β + n − b + c′ − c ≥ m − a′ + α , β + p − c.

It follows that s = max{q − x′ + α, β + p − c}

= max{n − a′ + a − b′ + α, β + p − c}

= n − a′ + a − b′ + α

= β + n − b + c′ − c

= max{m − a′ + α, β + n − b + c′ − c}

= max{m − a′ + α, β + v − z} = t.

Case 3: Assume that m − a ≥ n − b′ and n − b ≥ p − c′. The following inequalities hold: m − a′ + α ≥ β + p − c and m − a′ + α ≥ β + v − z. E. D. Schwab 239

It follows that s = max{q − x′ + α, β + p − c}

= max{m − a′ + α, β + p − c}

= m − a′ + α

= max{m − a′ + α, β + v − z} = t.

Case 4: Assume that m − a ≤ n − b′ and n − b ≤ p − c′. Then, n − a′ + a − b′ + α ≤ β + p − c and m − a′ + α ≤ β + p − c, and therefore s = max{q − x′ + α, β + p − c}

= max{n − a′ + a − b′ + α, β + p − c}

= β + p − c

= max{m − a′ + α, β + p − c}

= max{m − a′ + α, β + v − z} = t.

Thus we have proved associativity.

The set { } E(I) = (a, a, m) ∈ N3 a ≤ m and m ≠ 0 is the set of idempotents of I. For any two idempotents (a, a, m), (b, b, n) we have (a, a, m) · (b, b, n) = (b, b, n) · (a, a, m) and (a, a′, m) is the generalized inverse of (a′, a, m). So, the semigroup (I, ·) is an inverse semigroup. Since (a′, a, m)−1 · (a′, a, m) = (a, a, m) and (a′, a, m) · (a′, a, m)−1 = (a′, a′, m), it follows that   (a′, a, m)L (b′, b, n) if and only if a = b and m = n,   ′ ′ ′ ′ (a , a, m)R(b , b, n) if and only if a = b and m = n,  ′ ′  (a , a, m)D(b , b, n) if and only if m = n,  (a′, a, m)H (b′, b, n) if and only if a′ = b′, a = b and m = n. So, the inverse semigroup (I, ·) is aperiodic but not bisimple. Before proving the main theorem, a preliminary lemma from [5] is required (the assertions of the lemma are proved easily by induction). If S is a semigroup then s0 will denote the element 1 ∈ S1, where S1 is the semigroup obtained from S by adjoining an identity if necessary. 240 THEFREEMONOGENICINVERSESEMIGROUPANDTHEBICYCLICMULTIPLICATION

Lemma 2.1. Let S be an inverse semigroup and s ∈ S. (i) If 0 ≤ t ≤ r, then srs−tst = sr, (ii) If 0 ≤ t ≤ r, then srs−rst = srs−(r−t), (iii) If 0 ≤ t ≤ r, then sts−rsr = s−(r−t)sr, where s−p means (s−1)p = (sp)−1. Theorem 2.2. Let X = {x} be a singleton. The inverse semigroup (I, ·) together with the function f : X → I defined by f(x) = (0, 1, 1) is the free (monogenic) inverse semigroup on X.

Proof. Let S be an inverse semigroup and let g : X → S, say with g(x) = s. We define h : I → S by ′ h(a′, a, m) = s−a sms−(m−a). Obviously, h · f = g. It remains to check that h is a morphism and that h is uniquely determined. To see that h is a morphism, let α = (a′, a, m), β = (b′, b, n) ∈ I. Then h(α · β) = h((a′, a, m) · (b′, b, n))

= h((a′, a) ◦ (b′, b), max{(m, a) ◦ (b′, n)})

= h(a′ − a + max{a, b′}, max{a, b′} − b′ + b, max{m − a + max{a, b′},

max{a, b′} − b′ + n}) and h(α)h(β) = h(a′, a, m)h(b′, b, n)

′ ′ = s−a sms−(m−a)s−b sns−(n−b)

′ ′ = s−a sms−(m−a+b )sns−(n−b). Case 1: Assume that a ≥ b′ and m ≤ a − b′ + n. Then ′ ′ h(α · β) = h(a′, a − b′ + b, a − b′ + n) = s−a sa−b +ns−(n−b) and ′ ′ h(α)h(β) = s−a sms−(m−a+b )sns−(n−b)

′ ′ ′ ′ ′ ′ = s−a sa−b sm−a+b s−(m−a+b )sm−a+b sn−(m−a+b )s−(n−b)

′ ′ ′ ′ = s−a sa−b sm−a+b sn−(m−a+b )s−(n−b)

′ ′ = s−a sa−b +ns−(n−b).

Case 2: Assume that a ≥ b′ and m ≥ a − b′ + n. Then ′ ′ h(α · β) = h(a′, a − b′ + b, m) = s−a sms−(m−a+b −b) E. D. Schwab 241 and ′ ′ ′ ′ ′ ′ h(α)h(β) = s−a sms−(m−a+b )sns−(n−b) = s−a sa−b sm−a+b s−(m−a+b )sns−(n−b). Using Lemma 2.1 (ii) (with r = m − a + b′ and t = n), it follows ′ ′ ′ ′ ′ ′ h(α)h(β) = s−a sa−b sm−a+b s−(m−a+b −n)s−(n−b) = s−a sms−(m−a+b −b).

Case 3: Assume that a ≤ b′ and m ≤ a − b′ + n. Then ′ ′ h(α · β) = h(a′ − a + b′, b, n) = s−(a −a+b )sns−(n−b) and ′ ′ h(α)h(β) = s−a sms−(m−a+b )sns−(n−b)

′ ′ ′ ′ = s−a sms−(m−a+b )sm−a+b sn−(m−a+b )s−(n−b). Using Lemma 2.1 (iii) (with r = m − a + b′ and t = m), we obtain ′ ′ ′ ′ ′ ′ h(α)h(β) = s−a s−(−a+b )sm−a+b sn−(m−a+b )s−(n−b) = s−(a −a+b )sns−(n−b).

Case 4: Assume that a ≤ b′ and m ≥ a − b′ + n. Then ′ ′ h(α)h(β) = s−a sms−(m−a+b )sns−(n−b). By Lemma 2.1 (ii), sns−(n−b) = sns−nsb and therefore ′ ′ h(α)h(β) = s−a sms−(m−a+b )sns−nsb

′ ′ = s−a sm(s−1)m−a+b (s−1)−n(s−1)nsb. Using Lemma2.1 (i) (with r = m − a + b′ and t = n), we obtain ′ ′ h(α)h(β) = s−a sms−(m−a+b )sb. In addition, h(α · β) = h(a′ − a + b′, b, m − a + b′)

′ ′ ′ ′ = s−(a −a+b )sm−a+b s−(m−a+b −b)

′ ′ ′ ′ = s−a s−(−a+b )sm−a+b s−(m−a+b −b). Then, by Lemma 2.1 (iii) (with r = m − a + b′ and t = m), ′ ′ ′ ′ s−(a−b )sm−a+b = sms−(m−a+b )sm−a+b and therefore ′ ′ ′ ′ h(α · β) = s−a sms−(m−a+b )sm−a+b s−(m−a+b −b)

′ ′ ′ = s−a sm(s−1)m−a+b(s−1)−(m−a+b )(s−1)m−a+b −b. By Lemma 2.1 (ii) (with r = m − a + b′ and t = m − a + b′ − b), ′ ′ ′ ′ ′ (s−1)m−a+b (s−1)−(m−a+b )(s−1)m−a+b −b = (s−1)m−a+b (s−1)−b = s−(m−a+b )sb. 242 THEFREEMONOGENICINVERSESEMIGROUPANDTHEBICYCLICMULTIPLICATION

It follows that ′ ′ h(α · β) = s−a sms−(m−a+b )sb. We conclude that h is a morphism. Finally, we prove that h is unique. The following two equalities are proved easily by induction: (1, 0, 1)p = (p, 0, p) and (0, 1, 1)q = (0, q, q). If 0 ≤ a′, a ≤ m, then ′ (1, 0, 1)a ·(0, 1, 1)m · (1, 0, 1)m−a

= (a′, 0, a′) · (0, m, m) · (m − a, 0, m − a)

= ((a′, 0) ◦ (0, m), max{(a′, 0) ◦ (0, m)}) · (m − a, 0, m − a)

= (a′, m, m) · (m − a, 0, m − a)

= ((a′, m) ◦ (m − a, 0), max{(m, m) ◦ (m − a, m − a)})

= (a′, a, max{m, m}) = (a′, a, m). Since (1, 0, 1) = (0, 1, 1)−1, it follows that (0, 1, 1) generates the inverse semigroup I. This establishes that h is unique and completes the proof.  Remark 2.3. One of the isomorphic copies of the free monogenic inverse semi- group, constructed by Eberhart and Selden [1] (see also Petrich’s C4 [3, p. 399 and (iv) p. 406] where the last two coordinates of the triples are changed), is given by I = {(a′, a, m) ∈ N3 | a′, a ≤ m, m ≠ 0}, equipped with the product (a′, a, m) • (b′, b, n)

= (a′ − m + max{m, a + b′}, b − n + max{n, a + b′}, max{m, a + b′}

+ max{n, a + b′} − (a + b′)).

It is straightforward to check that our product “·” involving bicyclic multiplication is different from the Eberhart and Selden product. So, the inverse semigroup (I, ·) is a new isomorphic copy (on the same set I) of the free monogenic inverse semigroup.

Acknowledgements. The author would like to thank the referee for very valuable suggestions.

REFERENCES

[1] C. Eberhart and J. Selden, One-parameter inverse semigroups, Trans. Amer. Math. Soc. 168 (1972), 53–66. [2] L. M. Gluskin, Elementary generalized groups, Mat. Sb. N. S. 41 (83) (1957), 23–36. E. D. Schwab 243

[3] M. Petrich, Inverse semigroups, Pure and Applied Mathematics (New York), Wiley- Interscience Publication, John Wiley & Sons, Inc., New York, 1984, x+674 pp. [4] N. R. Reilly, Free inverse semigroups, in Algebraic theory of semigroups (Proc. Sixth Algebraic Conf., Szeged, 1976), pp. 479–508, Colloq. Math. Soc. János Bolyai, 20, North- Holland, Amsterdam-New York, 1979. [5] H. E. Scheiblich, A characterization of a free elementary inverse semigroup, Semigroup Forum 2 (1971), no. 1, 76–79.

E.D.SCHWAB,DEPT. OF MATH.SCIENCES,U. OF TEXAS AT EL PASO,EL PASO,TEXAS 79968, U.S.A. [email protected]