The bicyclic semigroup as the quotient inverse semigroup by any gauge inverse submonoid

: Every gauge inverse submonoid (including Jones-Lawson’s gauge inverse submonoid of the polycyclic monoid P n ) is a normal submonoid. In 2018, Alyamani and Gilbert introduced an equivalence relation on an inverse semigroup associated to a normal inverse subsemigroup. The corresponding quotient set leads to an ordered groupoid. In this note we shall show that this ordered groupoid is inductive if the normal inverse subsemigroup is a gauge inverse submonoid and the corresponding quotient inverse semigroup by any guage inverse submonoid is isomorphic either to the bicyclic semigroup or to the bicyclic semigroup with adjoined zero.


Introduction
An equivalence relation N on an inverse semigroup S associated to a normal inverse subsemigroup N is introduced in [1].Usually, it is not a congruence on S. Following [1] the quotient set S/ N (also denoted by S/ /N ) leads to an ordered groupoid [1,Theorem 3.6].If this ordered groupoid is inductive then the set of all morphisms, that is S/ /N , equipped with the "pseudoproduct" ⊗ ([3, page 112]) forms an inverse semigroup (see [3,Proposition 4.1.7(1)]), and we say, by abuse of language (since N is not necessary a congruence), that this inverse semigroup (S/ /N, ⊗) is the quotient inverse semigroup of S by the normal inverse subsemigroup N .
The gauge inverse monoid G M is a special submonoid of such a combinatorial bisimple (0-bisimple) inverse monoid S(M ) for which the submonoid M of right units is an -RILL monoid (see [5]).Any gauge inverse submonoid is normal ([5, Proposition 5.6]).Jones-Lawson's gauge inverse monoid is the gauge inverse submonoid (denoted by G n ) of the polycyclic monoid P n ([2, Section 3]).
The case of the polycyclic monoid P n is examined in Example 3.11 from [1].The conclusion of this examination is that P n / /G n is isomorphic to the Brandt semigroup on the set of non-negative integers.In fact the product "[(u, v)] Gn [(s, t)] Gn = [(u, t)] Gn " considered at the end of Section 3 in [1] is the composition of two morphisms (if it is defined) in the corresponding ordered groupoid and it is not the pseudoproduct ⊗ which defines the quotient inverse semigroup P n / /G n .
The aim of this note is to show that for any gauge inverse submonoid G M , the quotient inverse semigroup (S(M )/ /G M , ⊗) is isomorphic either to the bicyclic semigroup B or to the bicyclic semigroup with adjoined zero B 0 .
In the next section, we will survey the background results, particularly from [3] (Subsection 2.1), [1] (Subsection 2.2) and [5] (Subsection 2.3), needed to understand this paper.The symbol • is used only for composition (from right to left) of two morphisms.

Ordered groupoids
A groupoid G is a small category in which every morphism is an isomorphism, meaning that for any morphism f : where 1 X and 1 Y are the identity morphisms of X and Y , respectively.A groupoid G X is said to be connected simple system on the set X (or simplicial groupoid on X ) if the set of objects ObG X = X and there is exactly one morphism between any two objects.We call the groupoid G 0 X obtained from G X by adjoining an extra object 0 such that the set of morphisms from X to Y is empty if either X = 0, Y = 0 or X = 0, Y = 0 and it is a singleton if X = Y = 0, the connected simple system with adjoined 0.
A groupoid G is said to be ordered if the set of all morphisms Mor(G) of G is equipped with a partial order such that: An inverse semigroup S (i.e. a semigroup S in which every element s ∈ S has a unique inverse s −1 ∈ S in the sense that s = ss −1 s and s −1 = s −1 ss −1 ) can be considered as an ordered groupoid G(S) in which the set of objects is the set of idempotents E(S) of S, the set of morphisms from e to f is the set {s ∈ S|s −1 s = e and ss −1 = f } and the composition s • t of two morphisms s and t t is the usual product st in S (i.e., the composition is just the restriction of the multiplication of S to composable pairs).The partial order on the set of all morphisms of G(S) is the natural partial order ≤ on the inverse semigroup S, i.e. s ≤ t ⇔ s = ss −1 t (or equivalently s = ts −1 s).In the ordered groupoid G(S) the partially ordered set of identities forms a meet-semilattice.If S is the Brandt semigroup B ω whose set of elelements is with the multiplication defined by: ) is category isomorphic to the connected simple system with adjoined 0: G 0 ω .But G(B ω ) is an ordered groupoid and the order ≤ Bω on G(B ω ) (that is the natural partial order on B ω ) induces a partial order ≤ Bω on M or(G 0 ω ) given by: 1 0 ≤ Bω f for all f ∈ M or(G 0 ω ), and f ≤ Bω g iff f = g, otherwise.Note that G 0 ω (and G ω ) can be equipped as an ordered groupoid in many other way.Now, an ordered groupoid in which the set of identities forms a meet-semilattice (like in the case of the ordered groupoids G(S)) is called inductive.If f : X → Y and f : X → Y are two morphisms of an inductive groupoid G and 1 X ∧ 1 Y = 1 Z then the pseudoproduct ⊗:

Normal inverse subsemigroup and the corresponding ordered groupoid
An inverse subsemigroup N of an inverse semigroup S is called normal if E(S) = E(N ) and if s −1 N s ⊆ N for all s ∈ S. A normal inverse subsemigroup N of an inverse semigroup S together with the defining concepts (≤ and •) of the ordered groupoid G(S) determine a preorder ≤ N on S = MorG(S), as follows: Since ≤ N is a preorder on the set S then it defines an equivalence relation N on S by s N t ⇔ s ≤ N t and t ≤ N s, and a partial order on the set of equivalence classes S/ N .In [1] where a ∈ N such that a −1 a = tt −1 and aa −1 ≤ s −1 s; and [s] N N [t] N ⇔ s ≤ N t, is the partial order of G(S/ /N ).Now, if this ordered groupoid G(S/ /N ) is inductive then S/ /N = M or(G(S/ /N )) forms an inverse semigroup (S/ /N, ⊗) (where ⊗ is the pseudoproduct) called here the quotient inverse semigroup of S by the normal inverse subsemigroup N .

Gauge inverse submonoids
Following [5], a nontrivial right cancellative monoid M is a RILL monoid if 1 M is indecomposable and any two elements s, t ∈ M that admit a common left multiple admit a least common left multiple s ∨ t.In the RILL monoid M , we shall denote s t if t is a left multiple of s, t = rs, and by t s the "left quotient" r.Since M is right cancellative and 1 is indecomposable, the "right divisibility" relation is a partial order on M .A length function on the RILL monoid M is a monoid homomorphism : M → (N, +) such that −1 (0) = 1 M .A non-trivial monoid with a length function is atomic (every non-units element is a product of finitely many atoms).A length function is said to be normalized if (s) = 1 ⇔ s is an atom.An -RILL monoid is a RILL monoid equipped with a normalized length function .
If M is an -RILL monoid then the set is an inverse monoid (the inverse of (s, t) is (t, s); the element (s, t) is an idempotent if and only if s = t, and (1 M , 1 M ) is the identity element).The submonoid of S(M ): is the gauge inverse submonoid of S(M ) induced by the -RILL monoid M .This submonoid of S(M ) is a normal submonoid ([5, Proposition 5.6]).
In [5] the first example of a gauge inverse submonoid is the submonoid of idempotents E(B) of the bicyclic semigroup B. The bicyclic semigroup B is the monoid of all pairs of non-negative integers equipped with the multiplication defined by: In this paper (B 0 , •) denotes the bicyclic semigroup with adjoined zero 0.

Main results. The quotient inverse monoid S(M )/ /G M
Let M be an -RILL monoid and (S(M ), ) the corresponding inverse monoid.
Remark 3.2.The equivalence relation G M is not necessarily a congruence on S(M ).For example, if M is the multiplicative -RILL monoid of positive integers (Z + , •) ([5, Example 4.2]), where (1) = 0 and (n) =the total number of prime divisors of n counted with their multiplicities if n > 1, then S(Z + ) is the multiplicative analogue of the bicyclic semigroup: [n, m ] being the least common multiple of n and m .Now, if p and q are two distinct primes then (p, q) G M (p, q) and (p, q) G M (q, p) (since (p) = (q) = 1), but (p, q) • (p, q) = (p 2 , q 2 ) and (p, q) • (q, p) = (p, p), that is (p, q) • (p, q) G M (p, q) • (q, p).Thus G M is not a congruence on the multiplicative analogue of the bicyclic semigroup.
adjoining an extra element θ if necessary), together with the product defined by