ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.1112177 J. Algebra Comb. Discrete Appl. 9(2) • 123–131 Received: 8 October 2021 Accepted: 5 January 2022 Journal of Algebra Combinatorics Discrete Structures and Applications The bicyclic semigroup as the quotient inverse semigroup by any gauge inverse submonoid Research Article Emil Daniel Schwab Abstract: Every gauge inverse submonoid (including Jones-Lawson’s gauge inverse submonoid of the polycyclic monoid Pn) 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. 2010 MSC: 20M18, 20L05 Keywords: Inverse semigroup, Ordered groupoid, Gauge inverse submonoid, Bicyclic semigroup 1. 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 GM 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 Gn) of the polycyclic monoid Pn ([2, Section 3]). Emil Daniel Schwab; Department of Mathematical Sciences, University of Texas at El Paso, 500 W. University Ave, El Paso, Texas 79968-0514, USA (email: eschwab@utep.edu). 123 E. D. Schwab / J. Algebra Comb. Discrete Appl. 9(2) (2022) 123–131 The case of the polycyclic monoid Pn is examined in Example 3.11 from [1]. The conclusion of this examination is that Pn//Gn 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 Pn//Gn. The aim of this note is to show that for any gauge inverse submonoid GM, the quotient inverse semigroup (S(M)//GM,⊗) is isomorphic either to the bicyclic semigroup B or to the bicyclic semigroup with adjoined zero B0. 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. 2. Background. Ordered groupoids, normal inverse subsemi- groups and gauge inverse submonoids 2.1. Ordered groupoids A groupoid G is a small category in which every morphism is an isomorphism, meaning that for any morphism f : X → Y there is a morphism f−1 : Y → X such that f−1 ◦ f = 1X and f ◦ f−1 = 1Y , where 1X and 1Y are the identity morphisms of X and Y , respectively. A groupoid GX is said to be connected simple system on the set X (or simplicial groupoid on X) if the set of objects ObGX = X and there is exactly one morphism between any two objects. We call the groupoid G0X obtained from GX by adjoining an extra object 0 such that the set of morphisms from X to Y is empty if either X = 0,Y 6= 0 or X 6= 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: (O1) f � g implies f−1 � g−1; (O2) If f � g, f′ � g′ and f ◦f′ and g ◦g′ are defined then f ◦f′ � g ◦g′; (O3) If 1Z � 1X and f : X → Y then there exists a unique morphism f|Z : Z →• called the restriction of f to Z such that f|Z � f; (O4) If 1Z � 1Y and f : X → Y then there exists a unique morphism f|Z : •→ Z called the corestriction of f to Z such that f|Z � f; The axiom (O4) is a consequence of axioms (O1)− (O3). 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−1s and s−1 = s−1ss−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−1s = e and ss−1 = f} and the composition s◦ t of two morphisms s and t t−1t t−→ tt−1 = s−1s s−→ ss−1 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−1t (or equivalently s = ts−1s). In the ordered groupoid G(S) the partially ordered set of identities forms a meet-semilattice. If S is the Brandt semigroup Bω whose set of elelements is {(m,n) | m,n ∈ ω = {0,1,2, · · ·}}∪{0} with the multiplication defined by: (m,n) · (m′,n′) = { (m,n′) if n = m′ 0 if n 6= m′ and 0 · (m,n) = (m,n) ·0 = 0 ·0 = 0, 124 E. D. Schwab / J. Algebra Comb. Discrete Appl. 9(2) (2022) 123–131 then G(Bω) is category isomorphic to the connected simple system with adjoined 0: G0ω. But G(Bω) is an ordered groupoid and the order ≤Bω on G(Bω) (that is the natural partial order on Bω) induces a partial order ≤Bω on Mor(G0ω) given by: 10 ≤Bω f for all f ∈ Mor(G0ω), and f ≤Bω g iff f = g, otherwise. Note that G0ω (and Gω) 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 1X ∧1Y ′ = 1Z then the pseudoproduct ⊗: f ⊗f′ = f|Z ◦f′|Z defines a binary operation on the set Mor(G) such that (Mor(G),⊗) is an inverse semigroup ([3, Proposi- tion 4.1.7 (1)]). Note that if we denote this semigroup by S(G), then S(G(S)) = S ([3, Proposition 4.1.7 (3)]), G(S(G,�)) = (G,�) ([3, Proposition 4.1.7 (2)]), and S(G0ω,≤) ∼= Bω only if ≤ is the induced order ≤Bω on Mor(G0ω) considered above. 2.2. 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−1Ns ⊆ 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: s ≤N t ⇔ there exist two morphisms a,b of G(S) such that a,b ∈ N, the compositions a ◦ s and s ◦ b are both defined, and a◦s◦ b ≤ t. 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] this quotient set is denoted by S//N and the 'N- class of s ∈ S by [s]N. The equivalence relation 'N needs not be a congruence on S. However, the quotient set S//N leads us to an ordered groupoid G(S//N): the objects are the classes [e]N where e ∈ E(S), and Mor(G(S//N) = S//N with [s]N being a morphism from [s−1s]N to [ss−1]N. The composition of two morphisms [s]N ◦ [t]N (if [s−1s]N = [tt−1]N) is given by [s]N ◦ [t]N = [sat]N, where a ∈ N such that a−1a = tt−1 and aa−1 ≤ s−1s; 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 = Mor(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. 2.3. Gauge inverse submonoids Following [5], a nontrivial right cancellative monoid M is a RILL monoid if 1M 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 tBs 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) = 1M. 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 S(M) = { M ×M if Ms∩Mt 6= ∅ for any s,t ∈ M (M ×M)∪{θ} if there exist s,t ∈ M such that Ms∩Mt = ∅ 125 E. D. Schwab / J. Algebra Comb. Discrete Appl. 9(2) (2022) 123–131 (that is M ×M, adjoining an extra element θ if necessary), together with the product � defined by (s,t)� (s′, t′) = { (t∨s ′ Bt s, t∨s′ Bs′ t ′) if t and s′ admit a common left multiple θ otherwise and θ � (s,t) = (s,t)�θ = θ �θ = θ (if necessary), 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 (1M,1M) is the identity element). The submonoid of S(M): GM = { {(s,t) ∈ M ×M| `(s) = `(t)} if S(M) = M ×M {(s,t) ∈ M ×M| `(s) = `(t)}∪{θ} if S(M) = (M ×M)∪{θ} 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: (m,n) · (m′,n′) = { (m,n−m′ + n′) if n ≥ m′ (m−n + m′,n′) if n ≤ m′. In this paper (B0, ·) denotes the bicyclic semigroup with adjoined zero 0. 3. Main results. The quotient inverse monoid S(M)//GM Let M be an `-RILL monoid and (S(M),�) the corresponding inverse monoid. Proposition 3.1. The natural partial order ≤, the preorder ≤GM and the equivalence relation 'GM on S(M) are given by: (i) (s,t) ≤ (s′, t′) ⇔ s′ � s, t′ � t and sBs′ = t Bt′ ([4, Proposition 2.6 (1)]) (θ ≤ x for any x ∈ S(M) if S(M) = (M ×M)∪{θ}); (ii) (s,t) ≤GM (s′, t′) ⇔ there exists (p,q) ∈ S(M) such that `(p) = `(s), `(q) = `(t) and (p,q) ≤ (s′, t′) (θ ≤GM x for any x ∈ S(M) if S(M) = (M ×M)∪{θ}); (iii) (s,t) 'GM (s′, t′) ⇔ `(s) = `(s′) and `(t) = `(t′) (if S(M) = (M ×M)∪{θ} then the 'GM -class [θ]GM is a singleton). Proof. (i). We have (s,t) ≤ (s′, t′) ⇔ (s,t) = (s,t)� (t,s)� (s′, t′) ⇔ (s,t) = (s,s)� (s′, t′) ⇔ (s,t) = ( s∨s′ Bs s, s∨s′ Bs′ t′) ⇔ (s,t) = (s∨s′, s∨s′ Bs′ t′) ⇔ s′ � s and s Bs′ t′ = t ⇔ s′ � s, t′ � t and s Bs′ = t Bt′ . 126 E. D. Schwab / J. Algebra Comb. Discrete Appl. 9(2) (2022) 123–131 (ii). We have (s,t) ≤GM (s ′, t′) ⇔ there exist (p,u),(v,q) ∈ GM such that (p,u)−1 � (p,u) = (s,t)� (s,t)−1, (s,t)−1 � (s,t) = (v,q)� (v,q)−1 and (p,u)� (s,t)� (v,q) ≤ (s′, t′). Since (p,u)−1 � (p,u) = (u,u) and (s,t)� (s,t)−1 = (s,s), it follows u = s. Analogously, v = t. Now, we have: (p,u)� (s,t)� (v,q) = (p,s)� (s,t)� (t,q) = (p,q) and taking into account that (p,s),(t,q) ∈ GM we obtain: (s,t) ≤GM (s ′, t′) ⇔ there exist p,q ∈ M such that `(p) = `(s), `(q) = `(t) and (p,q) ≤ (s′, t′). (iii). The assertion follows from (i) and (ii). Remark 3.2. The equivalence relation 'GM 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: S(Z+) = Z+ ×Z+; (m,n) · (m′,n′) = ( [n,m′] n m, [n,m′] m′ n′), [n,m′] being the least common multiple of n and m′. Now, if p and q are two distinct primes then (p,q) 'GM (p,q) and (p,q) 'GM (q,p) (since `(p) = `(q) = 1), but (p,q) · (p,q) = (p2,q2) and (p,q) · (q,p) = (p,p), that is (p,q) ·(p,q) 6'GM (p,q) ·(q,p). Thus 'GM is not a congruence on the multiplicative analogue of the bicyclic semigroup. The 'GM -class [(s,t)]GM = {(u,v) ∈ S(M)| `(u) = `(s) and `(v) = `(t)} is a morphism in the ordered groupoid G(S(M)//GM) from [(t,t)]GM to [(s,s)]GM . If [(s,t)]GM and [(s′, t′)]GM are two morphisms of G(S(M)//GM) such that `(s′) = `(t) (that is [(s′,s′)]GM = [(t,t)]GM ), [(t′, t′)]GM [(s′,t′)]GM−→ [(s′,s′)]GM = [(t,t)]GM [(s,t)]GM−→ [(s,s)]GM , then the composition of these two morphisms, [(s,t)]GM ◦ [(s′, t′)]GM is given by [(s,t)]GM ◦ [(s ′, t′)]GM = [(s,t)� (a,b)� (s ′, t′)]GM , where (a,b) ∈ GM such that (a,b)−1 � (a,b) = (s′, t′) � (s′, t′)−1 and (a,b) � (a,b)−1 ≤ (s,t)−1 � (s,t). We choose (a,b) = (t,s′) which is an element of GM since `(t) = `(s′). Thus the composition [(s,t)]GM ◦ [(s′, t′)]GM in G(S(M)//GM) such that `(t) = `(s′) is given by: [(s,t)]GM ◦ [(s ′, t′)]GM = [(s,t)� (t,s ′)� (s′, t′)]GM = [(s,t ′)]GM . 127 E. D. Schwab / J. Algebra Comb. Discrete Appl. 9(2) (2022) 123–131 The ordering �GM of 'GM -classes in the ordered groupoid G(S(M)//GM) is given by: [(s,t)]GM �GM [(s ′, t′)]GM ⇔ there exists (p,q) ∈ [(s,t)]GM such that s′ � p, t′ � q and p Bs′ = q Bt′ . and [θ]GM �GM [x]GM for any morphism [x]GM of G(S(M)//GM) if S(M) = (M ×M)∪{θ}. Remark 3.3. The objects of G(S(M)//GM) other than [θ]GM (that is the 'GM -classes [(s,s)]GM) can be indexed by non-negative integers (namely [(s,s)]GM by `(s)), then the set of morphisms from m to n is a singleton (for any pair (m,n) of non-negative integers) and, it goes without saying the composition of two morphisms. It follows: Theorem 3.4. The (ordered) groupoid G(S(M)//GM) is category isomorphic either to the connected simple system GN (if S(M) = M ×M) or to the connected simple system with adjoined 0: G0N (if S(M) = M ×M ∪{θ}). Theorem 3.5. The ordered groupoid G(S(M)//GM) is inductive. Proof. It is straightforward to see that in the set of identities of G(S(M)//GM) we have: [(s,s)]GM �GM [(t,t)]GM ⇔ `(t) ≤ `(s). It follows that the partially ordered set of identities of G(S(M)//GM) forms a meet-semilattice: [(s,s)]GM ∧ [(t,t)]GM = { [(s,s)]GM if `(s) ≥ `(t) [(t,t)]GM if `(s) ≤ `(t) and [θ]GM ∧ [x]GM = [θ]GM for any identity morphism [x]GM of G(S(M)//GM) if S(M) = (M ×M)∪{θ}. Therefore the ordered groupoid G(S(M)//GM) is inductive. Theorem 3.6. The corresponding inverse semigroup (S(M)//GM,⊗) is isomorphic either to the bicyclic semigroup (B, ·) (if S(M) = M ×M) or to the bicyclic semigroup with adjoined zero (B0, ·) (if S(M) = (M ×M)∪{θ}). Proof. Let [(s,t)]GM , [(s ′, t′)]GM ∈ S(M)//GM. As morphisms of G(S(M)//GM), we have: [(s,t)]GM : [(t,t)]GM → [s,s]GM and [(s ′, t′)]GM : [(t ′, t′)]GM → [s ′,s′]GM . Since [(t,t)]GM ∧ [(s ′,s′)]GM = { [(t,t)]GM if `(t) ≥ `(s′) [(s′,s′)]GM if `(t) ≤ `(s′). we shall consider two cases: 128 E. D. Schwab / J. Algebra Comb. Discrete Appl. 9(2) (2022) 123–131 1) [(t,t)]GM ∧[(s′,s′)]GM = [(t,t)]GM . Then the restriction [(s,t)]GM |[(t,t)]GM of [(s,t)]GM to [(t,t)]GM is just [(s,t)]GM . The corestriction [(s ′, t′)]GM | [(t,t)]GM of [(s′, t′)]GM to [(t,t)]GM is the morphism [(t,y)]GM : [y,y]GM → [t,t]GM , where y ∈ M such that `(y) = `(t)− `(s′) + `(t′), since [(t,y)]GM �GM [(s′, t′)]GM . In this case, [(s,t)]GM ⊗ [(s ′, t′)]GM = [(s,t)]GM |[(t,t)]GM ◦ [(s ′, t′)]GM|[(t,t)]GM = [(s,t)]GM ◦ [(t,y)]GM = [(s,y)]GM . 2) [(t,t)]GM ∧ [(s′,s′)]GM = [(s′,s′)]GM . Then the restriction [(s,t)]GM |[(s′,s′)]GM of [(s,t)]GM to [(s′,s′)]GM is the morphism [(x,s ′)]GM : [(s ′,s′)]GM → [(x,x)]GM , where x ∈ M such that `(x) = `(s′)− `(t) + `(s) since [x,s′]GM �GM [(s,t)]GM . The corestriction [(s′, t′)]GM | [(s′,s′)]GM of [(s′, t′)]GM to [(s ′,s′)]GM is just [(s′, t′)]GM . So, in this case, the product [(s,t)]GM ⊗ [(s′, t′)]GM is given by: [(s,t)]GM ⊗ [(s ′, t′)]GM = [(s,t)]GM |[(s′,s′)]GM ◦ [(s ′, t′)]GM|[(s ′,s′)]GM = [(x,s′)]GM ◦ [(s ′, t′)]GM = [(x,t ′)]GM . (If S(M) = (M × M) ∪ {θ}) then it is straightforward to check that [θ]GM is the zero element of (S(M)//GM,⊗).) Now, a careful examination shows that ` : (S(M)//GM,⊗) → (B, ·) if S(M) = M ×M (` : (S(M)//GM,⊗) → (B0, ·) if S(M) = (M ×M)∪{θ}) defined by `([(s,t)]GM ) = (`(s),`(t)) (and `([θ]GM ) = 0 if S(M) = (M ×M)∪{θ}) is a monoid isomorphism. Remark 3.7. What is happening if the `-RILL monoid M is the additive monoid of non-negative integers ? (that is if the monoid (S(M),�) is the bicyclic semigroup B ?) The gauge inverse submonoid of B is the semilattice of idempotents E(B) ([5, Example 4.1]). It is straightforward to check that 'E(B) is the trivial relation (the equality) on B and of course B//E(B) = B (and G(B//E(B)) = G(B)). Now, since for any inverse semigroup S the relation 'E(S) is the trivial relation on S ([1, Proposition 3.4 (g)]), it follows that Corollary 3.8. The bicyclic semigroup is the only combinatorial bisimple inverse monoid for which the gauge inverse submonoid is the semilattice of idempotents. Remark 3.9. The ordered groupoid G(S(M)//GM) is isomorphic either to the ordered groupoid G(B) or to the ordered groupoid G(B0). Of course, the groupoids G(Pn//Gn) and G(Bω) are also isomorphic as two categories (since both are category isomorphic to the connected simple system with adjoined 0: G0ω), but they are not isomorphic as two ordered groupoids due to the two partial orders �Gn and ≤Bω on G(Pn//Gn) and G(Bω), respectively. 129 E. D. Schwab / J. Algebra Comb. Discrete Appl. 9(2) (2022) 123–131 4. Supplements. The quotient group S(M)/GM If ρ is a relation on an inverse semigroup S, the kernel kerρ is the set kerρ = {s ∈ S | sρe for some e ∈ E(S)}. If ρ is a group congruence on S then we agree to write S/kerρ for the quotient group S/ρ. In what follows assume that S(M) = M ×M (that is, Ms∩Mt 6= ∅ for any s,t ∈ M). We have: Proposition 4.1. The relation ≈M on S(M) defined by (x,y) ≈M (x′,y′) if and only if `(x)− `(y) = `(x′)− `(y′), is a group congruence on S(M). The gauge inverse submonoid GM is the kernel of ≈M, and it is the identity element of the quotient group S(M)/GM (= S(M)/ ≈M). This quotient group is isomorphic to the additive group of integers (Z,+). Proof. The relation ≈M is an equivalence relation on S(M). Obviously, GM is the kernel of ≈M . If (s,t) ≈M (s′, t′) and (u,v) ≈M (u′,v′), then (s,t) � (u,v) = (t∨u.t s, t∨u .u v), (s′, t′) � (u′,v′) = (t ′∨u′ .t′ s′, t ′∨u′ .u′ v′) and `( t∨u .t s)− `( t∨u .u v) = `(t∨u)− `(t) + `(s)− (`(t∨u)− `(u) + `(v)) = `(s)− `(t) + `(u)− `(v) = `(s′)− `(t′) + `(u′)− `(v′) = `( t′ ∨u′ .t′ s′)− `( t′ ∨u′ .u′ v′). It follows that ≈M is a congruence relation on S(M). The quotient monoid S(M)/ ≈M is again an inverse monoid. Since GM is the only idempotent of S(M)/ ≈M it follows that this inverse monoid is a group (the quotient group S(M)/GM). The map ` : S(M)/GM → Z defined by ([x,y]≈M ∈ S(M)/ ≈M) `([x,y]≈M ) = `(x)− `(y) is an isomorphism from the group S(M)/GM onto the additive group of integers (Z,+). Remark 4.2. It is straightforward to see that the kernel of 'GM is also the gauge inverse submonoid GM. However, the differences between the relations 'GM and ≈M are significant: (a) in general, the equivalence relation 'GM is not a congruence on S(M) (Remark 3.2), but ≈M is a group congruence on S(M); (b) the gauge inverse submonoid GM is not a 'GM -equivalence class in S(M), but it is an ≈M- equivalence class in S(M); (c) there is not a 'GM -equivalence class [(s,t)]GM such that E(S(M)) ⊆ [(s,t)]GM , but the ≈M- equivalence class GM contains the set of all idempotents of S(M); (d) the group S(M)/GM is equipped with the product � via the inverse monoid S(M); the product in the inverse monoid S(M)//GM is the pseudoproduct ⊗ via the inductive groupoid G(S(M)//GM); (e) the following inclusion holds: 'GM ⊂ ≈M . Acknowledgment: The author would like to thank the referee for helpful suggestions. 130 E. D. Schwab / J. Algebra Comb. Discrete Appl. 9(2) (2022) 123–131 References [1] N. Alyamani, N. D. Gilbert, Ordered groupoid quotients and congruences on inverse semigroups, Semigroup Forum 96 (2018) 506–522. [2] D. G. Jones, M. V. Lawson, Strong representations of the polycyclic inverse monoids: Cycles and atoms, Period. Math. Hung. 64 (2012) 54–87. [3] M. V. Lawson, Inverse semigroups: the theory of partial symmetries, World Scientific, Singapore (1998). [4] E. D. Schwab, Möbius monoids and their connection to inverse monoids, Semigroup Forum 90 (2015) 694–720. [5] E. D. Schwab, Gauge inverse monoids, Algebra Colloq. 27(2) (2020) 181–192. 131 https://doi.org/10.1007/s00233-017-9891-4 https://doi.org/10.1007/s00233-017-9891-4 https://doi.org/10.1007/s10998-012-9053-0 https://doi.org/10.1007/s10998-012-9053-0 https://doi.org/10.1142/3645 https://doi.org/10.1142/3645 https://doi.org/10.1007/s00233-014-9666-0 https://doi.org/10.1007/s00233-014-9666-0 https://doi.org/10.1142/S1005386720000152 Introduction Background. Ordered groupoids, normal inverse subsemigroups and gauge inverse submonoids Main results. The quotient inverse monoid S(M)//GM Supplements. The quotient group S(M)/GM References