1 Introduction

Subsemigroups of the bicyclic monoid L. Descalço and N. Ruˇskuc Mathematical Institute, University of St Andrews St Andrews KY16 9SS, Scotland Telefax: +44 1334 463748 E-mails: [email protected], [email protected] September 24, 2002 Abstract In this paper we give a description of all subsemigroups of the bicyclic monoid B. We show that there are essentially five different types of subsemigroups. One of them is the degenerate case, and the remaining four split in two groups of two, linked by the obvious anti-isomorphism of B. Each subsemigroup is characterized by a certain collection of parameters. Using our description, we determine the regular, simple and bisimple subsemigroups of B. Finally we describe algorithms for obtaining the parameters from the generating set. 2000 Mathematics Subject Classification: 20M10 1 Introduction The bicyclic monoid B is defined by the presentation hb; c j bc = 1i. A natural set of normal forms is fcibj : i; j ≥ 0g and we shall identify B with this set. The normal forms multiply according to the following rule: i−j+k l i j k l c b if j ≤ k c b c b = i j−k+l ( c b if j > k: The bicyclic monoid is one of the most fundamental semigroups. It is one of the main ingredients in the Bruck{Reilly extensions (see [7]), and also the basis of several general- izations; see [1],[2],[5],[6]. In [8, Sec 3.4] references are given to a number of applications of the bicyclic monoid to topics outside semigroup theory. The bicyclic monoid is known to have several remarkable properties, one of which is that it is completely determined by its lattice of subsemigroups; see [10] and [11]. Also, in [9] the authors study properties of a specific subsemigroup of B. Slightly surprisingly, there seems to be little other work in literature regarding the subsemigroups of B. 1 In this paper we give a description of all subsemigroups of B. We show that there are essentially five different types of subsemigroups. One of them is the degenerate case of subsets of fcibi : i ≥ 0g, and the remaining four split in two groups of two, linked by the obvious anti-isomorphism : cibj 7! cj bi of B. Each subsemigroup is characterized by a certain collection of parameters. We describe algorithms for obtaining these parameters from the generating set. b The paper is organized as follows. In Section 2 we define a series of distinguished subsets of B, which are then used as a kind of building blocks, and then we state our main theorem in Section 3. Section 4 contains the auxiliary results needed to prove the main theorem. In Sections 5 and 6 we respectively consider the two non-degenerate types of subsemigroups. In Section 7 we determine, using our description, the regular, simple and bisimple subsemigroups of B. Finally, Section 8 contains the algorithms for the computation of parameters. The classification of subsemigroups is used in the forth- coming paper [3] to investigate some properties of subsemigroups of B, such as finite presentability and automaticity. 2 Distinguished subsets In this section we introduce the notation we will need throughout the paper. In order to define subsets of the bicyclic monoid we find it convenient to represent B as an infinite square grid, as shown in Figure 1. We start by defining the functions Φ; Ψ; λ : B ! N0 by Φ(cibj ) = i, Ψ(cibj ) = j and λ(cibj) = jj − ij and by introducing some basic subsets of B: D = fcibi : i ≥ 0g − the diagonal ; U = fcibj : j > i ≥ 0g − the upper half; i j Rp = fc b : j ≥ p; i ≥ 0g − the right half plane (determined by p); i j Lp = fc b : 0 ≤ j < p; i ≥ 0g − the left strip (determined by p); i j Md = fc b : d j j − i; i; j ≥ 0g − the λ-multiples of d; for p ≥ 0 and d > 0. We now define the function : B ! B by cibj 7! cibj = cjbi. Geometrically is the reflection with respect to the main diagonal. So, for example, U is the lower half. Algebraically this function is anbanti-isomorphism (xy =dyx), as is easy to check. b By using the above basic sets and functions we now define some furtherb subsets of B that will be used in our description. For 0 ≤ q ≤ p ≤cm wbebdefine the triangle i j Tq;p = Lp \ Rq \ (U [ D) = fc b : q ≤ i ≤ j < pg; and the strips c i j Sq;p = Rp \ Rq \ Lp = fc b : q ≤ i < p; j ≥ pg; 0 i j Sq;p = Sq;p [ Tq;p = fc b : q ≤ i < p; j ≥ ig; c c i j Sq;p;m = Sq;p \ Rm = fc b : q ≤ i < p; j ≥ mg: 2 0 1 2 3 0 1 b b2 b3 ... 1 c cb cb2 cb3 ... 2 c2 c2b c2b2 c2b3 ... 3 c3 c3b c3b2 c3b3 ... .. Figure 1: The bicyclic monoid 3 1 4 4 1 1 U T S S' 1,4 1,4 1,4 3 4 Û Σ4 0 Figure 2: The upper and lower halves U; U, the triangle T1;4 and the strips S1;4; S1;4 b 4 Note that for q = p the above sets are empty. For i; m ≥ 0 and d > 0 we define the lines [ i j Λi = Ri \ Li+1 = fc b : j ≥ 0g; i j Λi;m;d = Λi \ Rm \ Md = fc b : d j j − i; j ≥ mg c and in general for I ⊆ f0; : : : ; m − 1g, i j ΛI;m;d = i2I Λi;m;d = fc b : i 2 I; d j j − i; j ≥ mg: For p ≥ 0, d > 0, r 2 [d] = Sf0; : : : ; d − 1g and P ⊆ [d] we define the squares i j Σp = Rp [ Rp = fc b : i; j ≥ pg; 1 1 \ p+r+ud p+r+vd Σp;d;r = Σpc\ ( Λp+r+ud) \ ( Λp+r+ud) = fc b : u; v ≥ 0g; u=0 u=0 [ p+r+ud[p+r+vd Σp;d;P = r2P Σp;d;r = fc b : r 2 P ; u; v ≥ 0g: Some of our subsetesS are illustrated in Figures 2 and 3. Finally, for X ⊆ S, we define ι(X) = min(Φ(X \ U)) (if X \ U 6= ;) and κ(X) = min(Ψ(X \ U)) (if X \ U 6= ;). b b 3 The main theorem We now state our main theorem, that will be proved in the following sections. Theorem 3.1 Let S be a subsemigroup of the bicyclic monoid. Then one of the following conditions holds: 1. The subsemigroup is a subset of the diagonal; S ⊆ D. 2. The subsemigroup is a union of a subset of a triangle, a subset of the diagonal above the triangle, a square below the triangle and some lines belonging to a strip determined by square and the triangle, or the reflection of this union with respect to the diagonal. Formally there exist q; p 2 N0 with q ≤ p, d 2 N, I ⊆ fq; : : : ; p−1g with q 2 I, P ⊆ f0; : : : ; d − 1g with 0 2 P , FD ⊆ D \ Lq, F ⊆ Tq;p such that S is of one of the following forms: (i) S = FD [ F [ ΛI;p;d [ Σp;d;P ; \ (ii) S = FD [ F [ ΛI;p;d [ Σp;d;P : N N 3. There exist d 2 b , I ⊆ 0, FD ⊆ D \ Lmin(I) and sets Si ⊆ Λi;i;d (i 2 I) such that S is of one of the following forms: (i) S = FD [ Si; i I [2 (ii) S = FD [ Si; i I [2 b 5 0 1 2 3 4 5 6 7 8 1 2 4 5 7 8 0 * * * 1 * * * 1 * * * 2 * * * 2 * * * 3 * * * 4 * * * 4 * * * 5 * * * 5 * * * 6 * * * 7 * * * 7 * * * 8 8 * * * * * * Figure 3: The λ-multiples of 3, M3, and the square Σ1;3;f0;1g 6 where each Si has the form Si = Fi [ Λi;mi;d for some mi 2 N0 and some finite set Fi, and I = I0 [ fr + ud : r 2 R; u 2 N0; r + ud ≥ Ng for some (possibly empty) R ⊆ f0; : : : ; d − 1g, some N 2 N0 and some finite set I0 ⊆ f0; : : : ; N − 1g. We start by observing that if S ⊆ D then there is nothing to describe because any i i idempotent c b is an identity for the square Σi below it. Condition 2. corresponds to subsemigroups having elements both above and below the diagonal; we call them two-sided subsemigroups. We observe that a subsemigroup defined by condition 2.(ii) is symmetric to the corresponding subsemigroup given by condition 2.(i) with respect to the diagonal, and so we can use the anti-isomorphism to obtain one from the other. Therefore we only need to consider subsemigroups that fall in one of these two categories. The description of two-sided subsemigroups is obtainedb in Section 5. We call upper subsemigroups those having all elements above the diagonal and lower subsemigroups those having all elements below the diagonal. Condition 3. corresponds to upper and lower subsemigroups. Again conditions 3.(i) and 3.(ii) give subsemigroups symmetric with respect to the diagonal and so only one of them will have to be considered. Upper subsemigroups are dealt with in Section 6. 4 Auxiliary results In this section we will prove some useful properties of the subsets defined in Section 2. Lemma 4.1 For any d 2 N the λ-multiples of d, Md, is a subsemigroup. i j k l i j k l i j−k+l Proof.

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