Two-Sided Expansions of Monoids 4.1

Home , Inverse element, Monoid (category theory)

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paper initiates such a study by showing that they have rich structure close to that of relatively free inverse monoids. Szendrei’s model [33] represents the elements of GR by pairs (A, g) with A ⊆ G being a finite subset containing 1 and g. It is this model of GR that found generalizations to e monoids. Applied to a monoid M, it produces a one-sided restriction monoid, Sz(M), the Szendrei expansion of M. In the group case, we have (eA, g)+ = (A, g)(A, g)−1 = (A, 1) and (A, g)∗ = (A, g)−1(A, g)=(g−1A, 1). In the monoid case, one puts (A, m)+ = (A, 1), which makes Sz(M) a one-sided restriction monoid. However, since monoids do not have involution, there is no similar way to extend (A, g)∗ to the monoid case. Szendrei’s model of GR appears in Exel’s work [11] as an inverse monoid being a quotient of a certain inverse monoid, S(G), given by generators and relations, determined by the property that any partiale action of G can be lifted to an action of S(G). Kellendonk and Lawson [24] completed Exel’s work by showing that S(G) is in fact isomorphic to GR. Due to the correspondence between partial actions and premorphisms, GR is uniquely e determined by its universal property: the map G → GR, g 7→ ({1,g},g), is a premorphism e universal amongst all premorphisms from G to inverse monoids. e In this paper, we take the universal property of GR, extended to the setting where M is a monoid, as the definition of an analogue of GR. In order to provide a unified treatment of e various classes of partial actions of monoids by partial bijections that extend partial actions of groups (or their relaxed version omittinge the requirement that the inverse element be preserved), we introduce the notions of a (· ,∗ ,+ , 1)-identity over X∗ and of a premorphism satisfying such an identity (see Section 3). The only requirement we need to impose for our arguments to go through, is that identities under consideration be satisfied by all monoid morphisms. We call such identities admissible. This leads to the notion of a class of partial actions of monoids defined by admissible identities. For example, the classes of all partial actions, of strong partial actions and of actions are defined by admissible identities. For a monoid M and a set, R, of admissible identities, we define FRR(M) to be the universal M-generated two-sided restriction monoid such that the inclusion map

ιFRR(M) : M → FRR(M) is a premorphism satisfying identities R. Our crucial observation is that the projection semilattice P (FRR(M)) turns out to be isomorphic to the idempotent semilattice E(FIR(M)) of the similarly deﬁned universal inverse monoid FIR(M). Com-

posing the inclusion premorphism ιFIR(M) : M → FIR(M) with the Munn representation

of FIR(M), one obtains a premorphism from M to TE(FIR(M)), the Munn inverse monoid of the semilattice E(FIR(M)). This data, via a partial action product construction, gives rise to the two-sided restriction monoid E(FIR(M)) ⋊ M (for its precise deﬁnition, see Subsection 6.2). Theorem 6.7, our main result, states that FRR(M) is isomorphic to E(FIR(M)) ⋊ M. In the case where R deﬁnes the class of monoid morphisms and M = A∗, we recover the result by Fountain, Gomes and Gould [14] on the structure of the free A-generated two-sided restriction monoid. Theorem 6.7 provides a precise sense of the main thesis of this paper that FRR(M), despite the absence of involution, has rich structure close to that of FIR(M). EXPANSIONS OF MONOIDS 3

We show that properties of the monoid M agree well with suitable properties of FRR(M). Thus M is left cancellative (resp. right cancellative or cancellative) if and only if FRR(M) is right ample (resp. left ample or ample), see Corollary 5.6. Further- more, M embeds into a group if and only if the canonical morphism FRR(M) → FIR(M) is injective, see Theorem 8.3. We pay special attention to the case where M is an inverse monoid and show that FIRs (M), the free inverse monoid over M with respect to strong premorphisms, is isomorphic to the Lawson-Margolis-Steinberg generalized prefix expansion M pr of M [27], see Theorem 8.8. In view of Theorem 6.7 and the known model for pr M [27], this leads to a model of FRRs (M), the free restriction monoid over M with respect to strong premorphisms. In the case where G is a group, FRRs (G) is isomorphic R to each of FIRs (G) and G , see Corollary 8.11. The paper is organized as follows. In Section 2 we recall definitions and facts on two- sided restriction semigroupse needed in this paper and prove Theorem 2.9. In Section 3 we define and discuss admissible (· ,∗ ,+ , 1)-identities over X∗ and the notion of a premorphism satisfying such an identity. In Section 4 we introduce the expansions FRR(M) and study their first properties. Further, in Section 5, we state the universal F -restriction property of FRR(M) which generalizes a result by Szendrei [33]. In Section 6 we introduce the inverse monoids FIR(M) and the partial action products E(FIR(M)) ⋊ M and then formulate Theorem 6.7. Section 7 is devoted to the proof of Theorem 6.7. Finally, in Section 8 we consider the special cases where M is embeddable into a group or is an inverse monoid.

2. Preliminaries 2.1. Restriction monoids. In this section we recall the deﬁnition and basic properties of two-sided restriction semigroups and monoids. Our presentation follows recent work [34, 35] on the subject. For more details, we refer the reader to [18, 22, 8]. A left restriction semigroup is an algebra (S; · ,+ ), where (S; · ) is a semigroup and + is a unary operation satisfying the following identities: (2.1) x+x = x, x+y+ = y+x+, (x+y)+ = x+y+, xy+ =(xy)+x. Dually, a right restriction semigroup is an algebra (S; · ,∗ ), where (S; · ) is a semigroup and ∗ is a unary operation satisfying the following identities: (2.2) xx∗ = x, x∗y∗ = y∗x∗, (xy∗)∗ = x∗y∗, x∗y = y(xy)∗. Left and right restriction semigroups are one-sided restriction semigroups. A two-sided restriction semigroup, orjust a restriction semigroup, is an algebra (S; · ,∗ ,+ ) of type (2, 1, 1), where (S; · ,+ ) is a left restriction semigroup, (S; · ,∗ ) is a right restriction semigroup and the following identities hold: (2.3) (x+)∗ = x+, (x∗)+ = x∗. A restriction semigroup possessing an identity element is called a restriction monoid. A restriction monoid is thus an algebra (S; · ,∗ ,+ , 1) of type (2, 1, 1, 0). Note that the axioms imply that, in a restriction monoid, 1∗ =1+ = 1. 4 GANNA KUDRYAVTSEVA

Morphisms and subalgebras of restriction semigroups and restriction monoids are taken with respect to the given signatures. It is immediate from the deﬁnition that restriction semigroups and restriction monoids form varieties of algebras. By F R(X) we denote the free restriction monoid over a set X. Let X be a set, S a restriction semigroup and ιS : X → S a (not necessarily injective) map. We say that S is X-generated via the map ιS if S is generated by ιS(X). Let S and T be restriction semigroups and ιS : X → S, ιT : X → T be maps. Let, further, ϕ: S → T be a morphism. We say that ϕ is X-canonical, or simply canonical, if ϕιS = ιT . Let S be a restriction semigroup. It follows from (2.3) that {s∗ : s ∈ S} = {s+ : s ∈ S}. This set, denoted by P (S), is closed with respect to the multiplication and is a semilattice with e ≤ f if and only if e = ef = fe and e ∧ f = ef. The top element of P (S) is 1 and e∗ = e+ = e holds for all e ∈ P (S). It is called the semilattice of projections of S and its elements are called projections. A projection is necessarily an idempotent, but a restriction semigroup may contain idempotents that are not projections. It follows that a restriction semigroup S satisfying P (S)= S is necessarily a semilattice. Since any semilattice S is a restriction semigroup with e∗ = e+ = e for all e ∈ S, restriction semigroups S satisfying S = P (S) are precisely semilattices. The opposite extreme are restriction semigroups S satisfying |P (S)| = 1. Then, necessarily, S is a monoid and P (S) = {1} so that s∗ = s+ = 1 holds for all s ∈ S. Since any monoid S is a restriction semigroup with s∗ = s+ = 1 for all s ∈ S, restriction semigroups S satisfying |P (S)| = 1 are precisely monoids. A monoid S, looked at as a restriction semigroup with |P (S)| = 1, is called a reduced restriction semigroup. The following equalities follow from the axioms and will be frequently used in the sequel: (2.4) es = s(es)∗ and se =(se)+s for all s ∈ S and e ∈ P (S);

(2.5) (st)∗ =(s∗t)∗, (st)+ =(st+)+ for all s, t ∈ S;

(2.6) (se)∗ =(s∗e)∗ = s∗e and (es)+ =(es+)+ = es+ for all s ∈ S and e ∈ P (S). In Section 7 we will need the following equalities that follow from (2.4) and (2.6): (2.7) ((se)+s)∗ = s∗e and (s(es)∗)+ = es+ for all s ∈ S and e ∈ P (S). The natural partial order ≤ on a restriction semigroup S is deﬁned, for s, t ∈ S, by s ≤ t if and only if there is e ∈ P (S) such that s = et. Elements s, t ∈ S are said to be compatible, denoted s ∼ t, if st∗ = ts∗ and t+s = s+t. The following properties related to the natural partial order and the compatibility relation will be used throughout the paper, often without reference. Lemma 2.1. Let S be a restriction semigroup and s, t ∈ S. Then: (1) s ≤ t if and only if s = tf for some f ∈ P (S); (2) s ≤ t if and only if s = ts∗ if and only if s = s+t; (3) s ≤ t implies su ≤ tu and us ≤ ut for all u ∈ S; EXPANSIONS OF MONOIDS 5

(4) s ≤ t implies s∗ ≤ t∗ and s+ ≤ t+; (5) s ≤ t implies s ∼ t; (6) if s, t ≤ u for some u ∈ S then s ∼ t; (7) if s ∼ t and s∗ = t∗ (or s+ = t+) then s = t. Restriction semigroups generalize inverse semigroups. Indeed, an inverse semgroup S can be endowed with the structure of a restriction semigroup by putting s∗ = s−1s and s+ = ss−1 for all s ∈ S. Note that, for S inverse, P (S) coincides with the semilattice E(S) of idempotents of S. As usual, inverse semigroups are considered in the signature (· ,−1) and inverse monoids in signature (· ,−1, 1). By FI(X) we denote the free inverse monoid over a set X. Since F R(X) is the free X-generated restriction monoid, there exist an X-canonical morphism (of restriction monoids) F R(X) → FI(X). As already mentioned, monoids and semilattices are basic examples of restriction semigroups, similar to the way in which groups and semilattices are basic examples of inverse semigroups. It is of fundamental importance that the Ehresmann-Namboripad-Schein theorem for inverse semigroups admits a natural extension to restriction semigroups, which states that the category of restriction semigroups is isomorphic to the category of induc- tive categories. Therefore, restriction semigroups are naturally arising generalizations of of inverse semigroups. However, in general, restriction semigroups are very far from being inverse, essentially because they do not have an involution. Let σ be the minimum congruence on a restriction semigroup S such that e σ f for all e, f ∈ P (S). Each of the following statements is equivalent to s σ t (see [9, Lemma 1.2]): (1) there is e ∈ P (S) such that es = et; (2) there is e ∈ P (S) such that se = te. Note that s ∼ t and, in particular, s ≤ t, imply s σ t, for all s, t ∈ S. The quotient S/σ is a reduced restriction semigroup and is the maximum reduced quotient of S. By σ♮ : S → S/σ we denote the moprhism induced by σ. We also put:

σ ♮ [t] = {s ∈ S : σ (s)= t}, t ∈ S/σ, and [s]σ = {v ∈ S : vσs}, s ∈ S. In general, if ρ is a congruence on an algebra S and s ∈ S, the congruence class of s is denoted by [s]ρ. A restriction semigroup S is called proper if the following two conditions hold: (1) for all s, t ∈ S : if s∗ = t∗ and s σ t then s = t; (2) for all s, t ∈ S : if s+ = t+ and s σ t then s = t. Note that a restriction semigroup is proper if and only if σ coincides with the compatibility relation ∼. Indeed, if S is proper and s σ t, we have t+sσs+t and (t+s)+ =(s+t)+. Thus t+s = s+t and similarly st∗ = ts∗ whence s ∼ t. It is easy to see and well known (see, e.g., [35, p. 282]) that in the case where S is an inverse semigroup, σ is the minimum group congruence on S, and S is proper if and only if it is E-unitary. Thus proper restriction semigroups generalize E-unitary inverse semigroups. 6 GANNA KUDRYAVTSEVA

A restriction semigroup S is called ample if for all s,t,u ∈ S: (2.8) su = tu ⇒ su+ = tu+ and us = ut ⇒ u∗s = u∗t. If only the first (resp., the second) of these two implications is required to hold, S is called left ample (resp. right ample). A semigroup S is called left (resp. right) cancellative provided that us = ut implies s = t (resp. su = tu implies s = t), for all s,t,u ∈ S. It is called cancellative if it is both left and right cancellative. If (A, ≤) is a poset and B ⊆ A, the order ideal generated by B is the set B↓ = {a ∈ A: a ≤ b for some b ∈ B}. If b ∈ A we write b↓ for {b}↓ and call this set the principal order ideal generated by b. 2.2. Premorphisms from a monoid to a restriction monoid. Let M be a monoid and T a restriction monoid. Definition 2.2. A premorphism from M to T is a map ϕ: M → T such that the following conditions hold: (PM1) ϕ(1) = 1; (PM2) ϕ(m)ϕ(n) ≤ ϕ(mn) for all m, n ∈ M. Clearly, the following are two equivalent forms of (PM2): (PM2′) ϕ(m)ϕ(n)= ϕ(mn)(ϕ(m)ϕ(n))∗ for all m, n ∈ M; (PM2′′) ϕ(m)ϕ(n)=(ϕ(m)ϕ(n))+ϕ(mn) for all m, n ∈ M. Because inverse monoids can be considered as a special case of restriction monoids, a premorphism from M to an inverse monoid arises as a special case of this definition. If the axiom (PM2) is replaced by a stronger axiom (PM3) ϕ(m)ϕ(n)= ϕ(mn) for all m, n ∈ M, the premorphism ϕ becomes, of course, a monoid morphism. Definition 2.3. A premorphism ϕ from a monoid M to a restriction monoid is called • left strong if ϕ(m)ϕ(n)= ϕ(m)+ϕ(mn) for all m, n ∈ M; • right strong if ϕ(m)ϕ(n)= ϕ(mn)ϕ(n)∗ for all m, n ∈ M; • strong if it is both left and right strong. Remark 2.4. Premorphisms from M to the symmetric inverse monoid I(X) have coun- terparts in terms of partial actions by partial bijections of M on X (see [25]), but in the present paper we choose to adhere to the language of premorphisms. For a comprehensive survey on partial actions, we refer the reader to [10]. 2.3. Proper restriction semigroups. We now recall the Cornock-Gould structure result on proper restriction semigroups [9]. We remark that in [9] a pair of partial actions, called a double action, satisfying certain compatibility conditions, was considered, and in [25] we reformulated this using one partial action by partial bijections. Here we restate the construction of [25] in terms of premorphisms. Let T be a monoid, Y =(Y, ∧) a semilattice EXPANSIONS OF MONOIDS 7

and assume that we are given a premorphism ϕ: T → I(Y ), t 7→ ϕt. We assume that for every t ∈ T the map ϕt satisﬁes the following axioms:

(A) dom ϕt and ran ϕt are order ideals of Y ; (B) ϕt : dom ϕt → ran ϕt is an order-isomorphism; (C) dom ϕt =6 ∅. We put: Y ⋊ϕ T = {(y, t) ∈ Y × T : y ∈ ran ϕt}.

If ϕ is understood, Y ⋊ϕ T is sometimes denoted simply by Y ⋊ T .

Remark 2.5. The notation Y ⋊ϕ T is new within the given context. In [9, 25] the notation M(T,Y ) is used instead. We opted to change the notation because, if T is a group, the construction agrees with the well-known construction of the partial transformation groupoid of a partial action (see, e.g., [30]), which, in turn, has deep ties with partial action cross product C∗-algebras via the work of Abadie [1], see [30].

∗ + The multiplication and the unary operations and on Y ⋊ϕ T are deﬁned by −1 (2.9) (x, s)(y, t)=(ϕs(ϕs (x) ∧ y), st);

∗ −1 + (2.10) (x, s) =(ϕs (x), 1), (x, s) =(x, 1). −1 −1 −1 Note that ϕs (x) ∧ y ≤ ϕs (x) so that ϕs (x) ∧ y ∈ dom ϕs, by axiom (A). Since −1 −1 y ∈ ran ϕt, we similarly have that ϕs (x) ∧ y ∈ ran ϕt. Hence ϕs (x) ∧ y ∈ dom ϕsϕt. −1 It now follows from ϕsϕt ≤ ϕst that ϕs (x) ∧ y ∈ dom ϕst, so that (2.9) makes sense. Furthermore, (2.10) makes sense since ϕ1 = idY . With respect to these operations, Y ⋊ϕ T is a restriction semigroup. It is proper and its semilattice of projections P (Y ⋊ϕ T )= {(y, 1): y ∈ Y } is isomorphic to Y via the map (y, 1) 7→ y, y ∈ Y . The minimum reduced congruence σ on Y ⋊ϕ T is given by (x, s) σ (y, t) if and only if s = t so that (Y ⋊ϕ T )/σ is isomorphic to T . Following the terminology of [32], it is natural to call Y ⋊ϕ T the semidirect type product of Y by T determined by the premorphism ϕ, or just the partial action product of Y by T where the partial action is determined by the premorphism ϕ. In the reverse direction, let S be a proper restriction semigroup. Deﬁnition 2.6. The underlying premorphism of S is the premorphism ϕ: S/σ → I(P (S)) given, for t ∈ S/σ, by σ ∗ (2.11) dom ϕt = {e ∈ P (S): there is s ∈ [t] such that e ≤ s } and

+ σ ∗ (2.12) ϕt(e)=(se) for any e ∈ dom ϕt and s ∈ [t] such that e ≤ s . It follows that, for all t ∈ S/σ, σ + (2.13) ran ϕt = {e ∈ P (S): there is s ∈ [t] such that e ≤ s } and 8 GANNA KUDRYAVTSEVA

−1 ∗ σ + (2.14) ϕt (e)=(es) for any e ∈ ran ϕt and s ∈ [t] such that e ≤ s . It can be verified (or see [9]) that ϕ is well defined and satisfies axioms (A), (B) and (C). Therefore, the partial action product P (S) ⋊ϕ S/σ can be formed and we have: σ + P (S) ⋊ϕ S/σ = {(e, t) ∈ P (S) × S/σ : there is s ∈ [t] such that e ≤ s }.

The operations on P (S) ⋊ϕ S/σ are: (e, u)(f, v)=(e(sf)+, uv), (e, u)∗ = ((es)∗, 1), (e, u)+ =(e, 1), where s ∈ [u]σ. The following theorem is due to Cornock and Gould [9].

Theorem 2.7. Let S be a proper restriction semigroup. Then the map f : S → P (S)⋊ϕS/σ given by f(s)=(s+, σ♮(s)) is an isomorphism. Remark 2.8. The fact that the underlying premorphism of a proper restriction semigroup has its image in an inverse semigroup reveals the connection between proper restriction semigroups and inverse semigroups. The main result of this paper, Theorem 6.7, makes this connection precise, for classes of proper restriction semigroups considered therein. We conclude this section with the following result. Theorem 2.9. Let S be a proper restriction semigroup. Then S is left ample (resp. right ample or ample) if and only if the monoid S/σ is right cancellative (resp. left cancellative or cancellative).

Proof. We put M = S/σ. Due to Theorem 2.7 we can assume that S is equal to P (S)⋊ϕ M where ϕ is the underlying premorphism of S. We ﬁrst prove the claim that S is left ample if and only if S/σ is right cancellative. We assume that P (S) ⋊ϕ M is left ample and that ac = bc in M. Let e ∈ ran ϕa, ′ ′ −1 −1 f ∈ ran ϕb and g ∈ ran ϕc. We put g = g ϕa (e)ϕb (f). Then g ∈ ran ϕc (by Axiom (A)), −1 −1 g ≤ ϕa (e)ϕb (f) so that g ∈ dom ϕa ∩ dom ϕb (by Axiom (B)). We show that ϕa(g) = −1 ϕb(g). Since g ∈ ran ϕc ∩dom ϕa, it follows that ϕc (g) ∈ dom ϕaϕc. But ϕaϕc ≤ ϕac, since −1 −1 −1 ϕ is a premorphism. Hence ϕc (g) ∈ dom ϕac and ϕa(g) = ϕaϕc(ϕc (g)) = ϕac(ϕc (g)). −1 −1 Similarly, ϕc (g) ∈ dom ϕbc and ϕb(g)= ϕbc(ϕc (g)). Hence, in view of ac = bc, we have ϕa(g)= ϕb(g), as desired. Note that (e, a), (f, b), (g,c) ∈ P (S) ⋊ϕ M and

(2.15) (e, a)(g,c)=(ϕa(g), ac)=(ϕb(g), bc)=(f, b)(g,c). In addition, + (2.16) (e, a)(g,c) =(e, a)(g, 1)=(ϕa(g), a) and, similarly, + (2.17) (f, b)(g,c) =(f, b)(g, 1)=(ϕb(g), b). It follows from (2.15), (2.16) and (2.17) that a = b, so that M is right cancellative. In the reverse direction, we assume that M is right cancellative. Let (e, a), (f, b), (g,c) ∈ P (S) ⋊ϕ M be such that (e, a)(g,c)=(f, b)(g,c). We calculate: −1 −1 (e, a)(g,c)=(ϕa(ϕa (e)g), ac) and (f, b)(g,c)=(ϕb(ϕb (f)g), bc). EXPANSIONS OF MONOIDS 9

−1 −1 The assumption implies that ϕa(ϕa (e)g) = ϕb(ϕb (f)g) and ac = bc. Since M is right cancellative, the latter equality yields a = b. But then: + −1 −1 + (e, a)(g,c) =(e, a)(g, 1)=(ϕa(ϕa (e)g), a)=(ϕb(ϕb (f)g), b)=(f, b)(g,c) .

It follows that P (S) ⋊ϕ M is left ample. We now prove the claim that S is right ample if and only if S/σ is left cancellative. We assume that P (S) ⋊ϕ M is right ample and that ca = cb in M. Note that if e ∈ ran ϕa and f ∈ ran ϕb then ef ∈ ran ϕa ∩ ran ϕb, by Axiom (A). Let e ∈ ran ϕa ∩ ran ϕb and f ∈ ran ϕc. Then (e, a), (e, b), (f,c) ∈ P (S) ⋊ϕ M and −1 −1 (2.18) (f,c)(e, a)=(ϕc(ϕc (f)e), ca)=(ϕc(ϕc (f)e), cb)=(f,c)(e, b). In addition, ∗ −1 −1 (2.19) (f,c) (e, a)=(ϕc (f), 1)(e, a)=(ϕc (f)e, a) and, similarly, ∗ −1 −1 (2.20) (f,c) (e, b)=(ϕc (f), 1)(e, b)=(ϕc (f)e, b). From (2.18) we have that (f,c)∗(e, a)=(f,c)∗(e, b). In view of (2.19) and (2.20), this yields that a = b, so that M is left cancellative. In the reverse direction, we assume that M is left cancellative. Let (e, a), (g, b), (f,c) ∈ P (S) ⋊ϕ M be such that (f,c)(e, a)=(f,c)(g, b). We calculate: −1 −1 (f,c)(e, a)=(ϕc(ϕc (f)e), ca) and (f,c)(g, b)=(ϕc(ϕc (f)g), cb). −1 −1 It follows that ϕc(ϕc (f)e) = ϕc(ϕc (f)g) and ca = cb. The first of these equalities −1 −1 simplifies to ϕc (f)e = ϕc (f)g and the second one implies that a = b. Furthermore, ∗ −1 ∗ similarly as in (2.19) and (2.20), we have (f,c) (e, a)=(ϕc (f)e, a) and (f,c) (g, b) = −1 ∗ ∗ ⋊ (ϕc (f)g, a). It follows that (f,c) (e, a)=(f,c) (g, b). Therefore, P (S) ϕ M is right ample. Combining the two results proved, it follows that S is ample if and only if M is cancellative, which finishes the proof. 3. Admissible identities Let X be a set and X∗ the free monoid over X. The elements of X∗ will be called variables, the empty word will be denoted by λ. Bya(· ,∗ ,+ , 1)-term over X∗ we mean an element of the free X∗-generated restriction monoid F R(X∗). We denote the identity element of F R(X∗) by 1 and remark that λ =6 1. ∗ If p ∈ F R(X ), we sometimes write p as p(u1,...,un) to indicate that the variables ∗ occurring in p are among u1,...,un ∈ X (see [5]). For a map f : X → M, where M is a monoid, we put f to be the free extension of f to X∗. Definition 3.1. bLet M be a monoid, S be a restriction monoid and f : X → M, γ : M → S ∗ be maps. The value of p(u1,...,un) ∈ F R(X ) on γ with respect to f is defined to be the element p(γ(f(u1)),...,γ(f(uk))) ∈ S which we denote by [p(u1,...,un)]γ,f . The followingb is immediate.b 10 GANNA KUDRYAVTSEVA

Lemma 3.2. Let M be a monoid, S1, S2 be restriction monoids, γ : M → S1 be a map ∗ and g : S1 → S2 be a morphism. Then for any p ∈ F R(X ) we have:

(3.1) [p]gγ,f = g([p]γ,f ). Deﬁnition 3.3. A (· ,∗ ,+ , 1)-identity over X∗ is an equality of the form

(3.2) p(u1,...,un)= q(u1,...,un) ∗ where p(u1,...,un), q(u1,...,un) ∈ F R(X ). In this paper we consider only (· ,∗ ,+ , 1)-identities over X∗, and, for brevity, we refer to them as (· ,∗ ,+ , 1)-identities, or simply identities. Deﬁnition 3.4. Let γ : M → S be a map from a monoid M to a restriction monoid S. We say that γ satisﬁes identity (3.2) if for every map f : X → M the equality

[p(u1,...,uk)]γ,f = [q(u1,...,uk)]γ,f holds in S. We say that γ satisfies a set R of identities if it satisfies each identity in R. Definition 3.5. Let R be a set of (· ,∗ ,+ , 1)-identities over X∗, and define V (R) to be the class of all premorphisms satisfying R. A class V of premorphisms from monoids to restriction monoids is an equational class of premorphisms (or simply an equational class), if V = V (R) for some set R of (· ,∗ ,+ , 1)-identities over X∗.

∗ + ∗ Definition 3.6. Two sets of (· , , , 1)-identities over X , R1 and R2, are called equivalent if V (R1)= V (R2). The class PM of all premorphisms from monoids to restriction monoids is the biggest equaitonal class of premorphisms, i.e., PM = V (∅). Clearly, V ⊆PM for any equational class of premorphisms V. If R′ consists of identities satisfied by all premorphisms then V (R) = V (R ∪ R′) = V (R \ R′) for any set R of identities. In particular, PM = V (R′) for any set R′ of identities satisfied by all premorphisms, for example, for ′ ∗ R = {λ =1, x1 · x2 = x1x2 · (x1 · x2) }. The following is a central definition. Definition 3.7. A (· ,∗ ,+ , 1)-identity over X∗ will be called admissible if all monoid morphisms satisfy this identity. We give some examples of sets of admissible identities and equational classes of premorphisms they define. Example 3.8. The set ∅ defines the class PM of all premorphisms. Furthermore, the sets

(3.3) Rm = {x1 · x2 = x1x2};

+ (3.4) Rls = {x1 · x2 = x1 · x1x2}; EXPANSIONS OF MONOIDS 11

∗ (3.5) Rrs = {x1 · x2 = x1x2 · x2};

(3.6) Rs = Rls ∪ Rrs

consist of admissible identities. The class V (Rm), denoted also by M, is the class of all monoid morphisms; the classes V (Rls), V (Rrs) and V (Rs) are the classes of all left strong premorphisms, right strong premorphisms, and premorphisms, respectively. We note that a set R of (· ,∗ ,+ , 1)-identities over X∗ consists of admissible identities if and only if M ⊆ V (R) ⊆ PM.

Remark 3.9. We are mostly interested in the cases R = ∅, R = Rm and R = Rs. Developing the theory for an arbitrary set R of admissible identities, we aim not only to extend the generality, but also to provide a uniﬁed and thus conceptually simpler treatment of the special cases of main interest.

∗ Observe that, applying (2.4), any p ∈ F R(X ) can be written as p = u1 · ... · uk · e, ∗ ∗ where k ≥ 1, ui ∈ X , 1 ≤ i ≤ k, and e ∈ P (F R(X )). It follows that an identity p = q, where p, q ∈ F R(X∗), can be rewritten as

(3.7) u1 · ... · uk · e = v1 · ... · vn · f,

∗ ∗ where k, n ≥ 1, u1,...,uk, v1,...,vn ∈ X and e, f ∈ P (F R(X )). By definition, a ∗ premorphism γ : M → S satisfies the identities u1 · ... · uk =(u1 ...uk) · (u1 · ... · uk) and ∗ v1 · ... · vn = (v1 ...vn) · (v1 · ... · vn) . It follows that γ satisfies the identity (3.7) if and only if it satisfies the identity

∗ ∗ (3.8) (u1 ...uk) · (u1 · ... · uk) · e =(v1 ...vn) · (v1 · ... · vn) · f. Therefore, for the identity p = q there is an equivalent identity of the form (3.9) u · e = v · f where u, v ∈ X∗ and e, f ∈ P (F R(X∗)). Assume that p = q is an admissible identity. Since it is satisﬁed by any monoid morphism, so does the identity u · e = v · f just found. In particular, it is satisﬁed by the ∗ morphism idX∗ . Thus u = v holds in X . It follows that the identity p = q is equivalent to an identity of the form (3.10) u · e = u · f where u ∈ X∗ and e, f ∈ P (F R(X∗)). We have proved the following. Proposition 3.10. Let R be a set of admissible identities. Then it is equivalent to another set, R′, of admissible identities that contains only identities of the form (3.10).

′ ′ ∗ ∗ For example, V (Rrs)= V (Rrs) where Rrs = {x1x2 · (x1 · x2) = x1x2 · x2}. 12 GANNA KUDRYAVTSEVA

4. Two-sided expansions of monoids 4.1. The expansions and the universal property. For a monoid M, we put ⌊M⌋ = {⌊m⌋: m ∈ M}. The free restriction monoid F R⌊M⌋ is M-generated via the map ιF R⌊M⌋ : M → F R⌊M⌋, m 7→ ⌊m⌋. Deﬁnition 4.1. Let M be a monoid. • We deﬁne FR(M) to be the quotient of F R⌊M⌋ by the congruence δ generated by the relations: e (4.1) ⌊1⌋ = 1; (4.2) ⌊m⌋⌊n⌋ = ⌊mn⌋(⌊m⌋⌊n⌋)∗, m, n ∈ M.

These relations say that the map ιFR(M) : M → FR(M), m 7→ ⌊m⌋, is a premorphism. We call FR(M) the the free restriction monoid over M with respect to premorphisms. ∗ + ∗ • Let R be a set of admissible (· , , , 1)-identites over X . We deﬁne FRR(M) to be the quotient of FR(M) by the congruence δR generated by the relations:

[p]ιFR(M),f = [q]ιFR(M),f , where f runs through all maps X → M and p = q runs through R. We call FRR(M) the the free restriction monoid over M with respect to R.

Note that FR∅(M) = FR(M). The deﬁnition of FRR(M) says that the map

ιFRR(M) : M → FRR(M), m 7→ ⌊m⌋, belongs to V (R). Moreover, FRR(M) is M- ♮ generated via the map ιFRR(M) and the projection morphism δR : FR(M) → FRR(M) is M-canonical. Deﬁnition 4.1 is equivalent to the following universal property of FRR(M). Proposition 4.2. Let R be a set of admissible identities and M be a monoid. Then for any premorphism λ: M → S, where S is a restriction monoid, satisfying λ ∈ V (R), there

is a morphism η : FRR(M) → S of restriction monoids such that λ = ηιFRR(M).

Remark 4.3. Let λ = idM . Then λ ∈ M ⊆ V (R). Proposition 4.2 implies that the map ⌊m⌋ 7→ m, m ∈ M, extends to an M-canonical morphism pM : FRR(M) → M. We record an immediate consequence of Remark 4.3.

Corollary 4.4. The premorphism ιFRR(M) is injective.

Proposition 4.5. Let R be a set of admissible identities and α: M1 → M2 a monoid morphism. Then there is a morphism α: FRR(M1) → FRR(M2) of restriction monoids such that the following diagram commutes: e αe FRR(M1) FRR(M2)

pM1 pM2

α M1 M2 EXPANSIONS OF MONOIDS 13

′ ′ Proof. Let α = ιFRR(M2)α. Then α ∈ V (R). By Proposition 4.2 it extends to a morphism α: FRR(M1) → FRR(M2). That the diagram above commutes is immediate from the construction of α. e Remark 4.6. Usinge categorical language, for any set R of admissible identities, we have constructed a functor, FRR, from the category of monoids to the category of restriction monoids, given on objects by M 7→ FRR(M) and on morphisms by α 7→ α˜. Moreover, there is a natural transformation, p, from FRR to the identity functor on the category of monoids, whose component at a monoid M, given by the map pM : FRR(M) → M, is surjective. It follows that the functor FRR is an expansion in the sense of [2]. Example 4.7.

• Let R = Rm (see (3.3)). We call FRRm (M) the free restriction monoid over M

(with respect to morphisms). The deﬁning relations for FRRm (M) are (4.1) and (4.3) ⌊m⌋⌊n⌋ = ⌊mn⌋, m, n ∈ M.

• Let R = Rls (see (3.4)). We call FRRls (M) the free restriction monoid over M with respect to left strong premorphisms. Its deﬁning relations are (4.1) and (4.4) ⌊m⌋⌊n⌋ = ⌊m⌋+⌊mn⌋, m, n ∈ M.

• Let R = Rrs (see (3.5)). We call FRRrs (M) the free restriction monoid over M with respect to right strong premorphisms. Its deﬁning relations are (4.1) and (4.5) ⌊m⌋⌊n⌋ = ⌊mn⌋⌊n⌋∗, m, n ∈ M.

• Let now R = Rs (see (3.6)). We call FRRs (M) the free restriction monoid over

M with respect to strong premorphisms. The deﬁning relations for FRRs (M) are (4.1), (4.4) and (4.5).

∗ ∗ Example 4.8. In the case where M = A is a free A-generated monoid, FRRm (A ) is isomorphic to the free restriction monoid F R(A). The following diagram illustrates the connection between the deﬁned expansions. Arrows represent components at M (which are all surjective) of the canonical natural transforma- tions between the expansions (looked at as functors). FR(M)

M M FRRls ( ) FRRrs ( )

M FRRs ( )

M FRRm ( ) 14 GANNA KUDRYAVTSEVA

5. F -restriction monoids and their associated premorphisms 5.1. The Munn representation of a restriction semigroup. Recall that the Munn semigroup TY of a semilattice Y is the semigroup of all order-isomorphisms between principal order ideals of Y under composition. This is an inverse semigroup contained in I(Y ). If Y has a top element, TY is a monoid whose identity is idY , the identity map on Y . The Munn representation of a restriction semigroup S is a morphism α: S → TP (S), s 7→ αs, given, for each s ∈ S, by ∗ ↓ + (5.1) dom αs =(s ) and αs(e)=(se) , e ∈ dom αs. It follows that + ↓ −1 ∗ (5.2) ran αs =(s ) and αs (e)=(es) , e ∈ ran αs. −1 −1 −1 If S is an inverse semigroup, αs(e)= ses for all e ∈ dom αs and αs (e)= αs−1 (e)= s es for all e ∈ ran αs. 5.2. The two premorphisms underlying an F -restriction monoid. A restriction semigroup is called F -restriction if every σ-class has a maximum element. It is easy to check (or see [25, Lemma 5]) that an F -restriction semigroup is necessarily a monoid with the identity element being the maximum projection, and is proper. Let S be an F -restriction monoid. We put T = S/σ. Inspired by similar considerations for F -inverse monoids [24], we introduce the map σ (5.3) τS : T → S given by τS(t) = max [t] , t ∈ T.

Lemma 5.1. The map τS is a premorphism.

Proof. It is immediate that τS(1) = 1. Let t1, t2 ∈ T . Since τS(t1)τS(t2) σ τS (t1t2), we have τS(t1)τS(t2) ≤ τS(t1t2). Throughout this section, S is an F -restriction monoid. Recall that ϕ is the underlying premorphism of S (see Deﬁnition 2.6). To simplify notation, we write τ for τS. Due to the fact that S is F -restriction, for each t ∈ T , (2.11), (2.12), (2.13) and (2.14) simplify to: ∗ ↓ + ∗ (5.4) dom ϕt =(τ(t) ) and ϕt(e)=(τ(t)e) , e ≤ τ(t) ;

−1 + ↓ −1 ∗ + (5.5) dom ϕt = ran ϕt =(τ(t) ) and ϕt (e)=(eτ(t)) , e ≤ τ(t) .

It follows that the image of ϕ is contained in TP (S) and, moreover, ϕ = ατ. Proposition 5.2. Let S be an F -restriction monoid, T = S/σ, ϕ the underlying premorphism of S and τ the premorphism given in (5.3). Then ϕ satisﬁes an admissible identity if and only if τ satisﬁes the same identity. Proof. Let p ∈ F R(X∗) and let g : X → T be a map. It follows from (5.1) and (5.2) that ∗ ↓ + ↓ (5.6) dom([p]ϕ,g) = ([p]τ,g) , ran([p]ϕ,g) = ([p]τ,g) and

+ ∗ −1 ∗ + (5.7) [p]ϕ,g(e) = ([p]τ,ge) , e ≤ [p]τ,g;[p]ϕ,g(e)=(e[p]τ,g) , e ≤ [p]τ,g. EXPANSIONS OF MONOIDS 15

Let p = q be an admissible identity. Since ϕ = ατ and α is a morphism, it folows, in view of Lemma 3.2, that [p]τ,g = [q]τ,g implies [p]ϕ,g = [q]ϕ,g. Suppose that [p]ϕ,g = [q]ϕ,g. Since the identity p = q is satisfied by idT , we have ♮ ♮ σ ([p]τ,g) = [p]idT ,g = [q]idT ,g = σ ([q]τ,g), where for the first and the third equalities we used Lemma 3.2 taking into account that ♮ ♮ idT = σ τ with σ being a morphism. Hence [p]τ,g σ [q]τ,g. Since, in addition, (5.6) implies ∗ ∗ that [p]τ,g = [q]τ,g and since S is proper, we obtain [p]τ,g = [q]τ,g. Let S be an F -restriction monoid. It is called left extra F -restriction (resp. right extra F - restriction or extra F -restriction) [25] provided that the underlying premorphism ϕS is left strong (resp. right strong or strong). It is called ultra F -restriction [25] (or perfect [23]) if ϕS is a morphism. Proposition 5.2 tells us that all these classes can be equivalently defined by the respective properties of the premorphism τS .

5.3. Universal F -restriction property. Lemma 5.3. Let M be a monoid and R a set of admissible identities. Then the congruence σ on FRR(M) coincides with ker(pM ). Consequently, FRR(M)/σ ≃ M.

Proof. Let a, b ∈ FRR(M) be such that pM (a) = pM (b). Applying (2.4), we rewrite the elements a and b as a = ⌊u⌋e and b = ⌊v⌋f where u, v ∈ M and e, f ∈ P (FRR(M)). Then u = pM (a)= pM (b)= v. Hence ⌊u⌋ef ≤ a, b which implies a σ b.

We can now state the universal F -restriction property of FRR(M). Corollary 8.11 shows that it extends a result by Szendrei [33]. Proposition 5.4. Let R be a set of admissible identities and M be a monoid.

(1) The restriction monoid FRR(M) is F -restriction and τFRR(M) = ιFRR(M). In particular, FRR(M) is proper. (2) Let S be an F -restriction monoid such that S/σ ≃ M and assume that τS ∈ V (R). Then there is a morphism of restriction monoids η : FRR(M) → S such that ♮ η⌊m⌋ = τS (m) for all m ∈ M and pM = σSη. Proof. (1) This is immediate. (2) This follows from Proposition 4.2.

Remark 5.5. Proposition 5.4 shows that the functor FRR from the category of monoids to the category of F -restriction monoids whose underlying premorphism belongs to V (R) is a left adjoint to the functor in the reverse direction that takes S to S/σ. The following result is a consequence of Theorem 2.9 and Proposition 5.4(1). Corollary 5.6. Let M be a monoid and R a set of admissible identities. Then M is left cancellative (resp. right cancellative, cancellative) if and only if FRR(M) is right ample (reps. left ample, ample). 16 GANNA KUDRYAVTSEVA

6. The main result 6.1. Free inverse monoids over M. For a monoid M, we put [M] = {[m]: m ∈ M}. The free inverse monoid FI[M] is M-generated via the map ιFI[M] : M → FI[M], m 7→ [m]. The following definition is analogous to Definition 4.1. Definition 6.1. Let M be a monoid. • We define FI(M) to be the quotient of FI[M] by the congruence γ generated by the relations: e (6.1) [1] = 1;

(6.2) [m][n] = [mn]([m][n])∗, m, n ∈ M.

The deﬁning relations say that the map ιFI(M) : M → FI(M), m 7→ [m], is a premorphism. We call FI(M) the free inverse monoid over M with respect to premorphisms. • Let R be a set of admissible identities and M be a monoid. We deﬁne FIR(M) to be the quotient of FI(M) by the congruence γR generated by the relations:

[p]ιFI(M),f = [q]ιFI(M),f , where f runs through all maps X → M and p = q runs through R. We call FIR(M) the free inverse monoid over M with respect to R. Standard arguments show that an analogue of Proposition 4.2 holds true, that is, the

map ιFIR(M) : M → FIR(M), m 7→ [m], is a premorphism satisfying identities R that is universal amongst all such premorphisms from M to inverse monoids. Observe that FI(M)= FI∅(M). We highlight several more examples of interest:

• FIRm (M), the free inverse monoid over M (with respect to morphisms);

• FIRls (M), the free inverse monoid over M with respect to left strong premorphisms.

• FIRrs (M), the free inverse monoid over M with respect to right strong premorphisms.

• FIRs (M), the free inverse monoid over M with respect to strong premorphisms. ∗ Example 6.2. FIRm (A ) is isomorphic to the free inverse monoid FI(A) over A.

Remark 6.3. The premorphism ιFIR(M) is not injective in general. For example, an easy calculation shows that if M is a rectangular band with the identity element added then

FIRm (M) is two-element.

Remark 6.4. It is immediate by the deﬁnition that if M is an inverse monoid, FIRm (M) is isomorhic to M. Thus one can not expect FIR(M) to be in general F -inverse or E-unitary. There is no analogue of Proposition 4.5 either. (The diﬀerence with the respective property of FRR(M) is essentially because the maximum reduced image of the inverse monoid FIR(M) is a group, and thus can not be just monoid M, as it is for FRR(M).) In particular, the functor FIR is not an expansion. EXPANSIONS OF MONOIDS 17

6.2. The partial action product E(FIR(M)) ⋊ M and the main result. Let R be ∗ −1 a set of admissible identities. Since FIR(M) is a restriction monoid (with a = a a and + −1 a = aa ) and ιFIR(M) ∈ V (R), Proposition 4.2 implies that there is an M-canonical morphism

(6.3) ψR : FRR(M) → FIR(M).

Note that ψR does not need to be surjective. Wondering if there is a closer connection between FRR(M) and FIR(M) we make the following crucial observations. Let α: FIR(M) → I(E(FIR(M))) be the Munn representation of FIR(M) (see Sub-

section 5.1). We put β = αιFIR(M) : M → I(E(FIR(M))), m 7→ βm. We have: ∗ ↓ + ↓ dom βm = ([m] ) , ran βm = ([m] ) ; −1 + ∗ βm(e) = [m]e[m] = ([m]e) , e ≤ [m] ; −1 −1 ∗ + βm (e) = [m] e[m]=(e[m]) , e ≤ [m] . Lemma 6.5. The map β is a premorphism and β ∈ V (R). Moreover, β satisﬁes conditions (A), (B), (C) of Subsection 2.3 for forming the partial action product E(FIR(M)) ⋊β M.

Proof. The ﬁrst assertion holds in view of Lemma 3.2 because β = αιFIR(M) with α being a morphism and ιFIR(M) ∈ V (R). The second assertion is immediate. In the sequel, to prevent overload of the notation, we suppress the index β of E(FIR(M)) ⋊β M. We have: + E(FIR(M)) ⋊ M = {(e, m) ∈ E(FIR(M)) × M : e ≤ [m] }.

The identity element of E(FIR(M)) ⋊ M is (1, 1) and the rest of the operations are: (6.4) (e, m)(f, n)=(e([m]f)+, mn), (e, m)∗ = ((e[m])∗, 1), (e, m)+ =(e, 1). Proposition 6.6.

(1) Let (e, m), (f, n) ∈ E(FIR(M)) ⋊ M. Then (e, m) σ (f, n) if and only if m = n. + Consequently, E(FIR(M)) ⋊ M is F -restriction with max[(e, m)]σ = ([m] , m). + (2) The map τR : M → E(FIR(M)) ⋊ M, given by τR(m) = ([m] , m), is a premorphism and τR ∈ V (R).

Proof. (1) Since E(FIR(M)) ⋊ M is proper, σ coincides with the compatibility relation ∼. We thus have (e, m) σ (f, n) if and only if (6.5) (e, m)(f, n)∗ =(f, n)(e, m)∗ and (f, n)+(e, m)=(e, m)+(f, n). The left-hand side of the first of these equalities equals (e, m)(f, n)∗ =(e, m)((f[n])∗, 1)=(e([m](f[n])∗)+, m). Similarly, the right-hand side equals (f, n)(e, m)∗ = (f([n](e[m])∗)+, n). Thus if the first equality of (6.5) holds then m = n. Conversely, if m = n then f([n](e[m])∗)+ = f([m](e[m])∗)+ = f(e[m])∗ = (e[m]f)∗ and, similarly, e([m](f[n])∗)+ = (e[n]f)∗ = (e[m]f)∗, so that the first equality of (6.5) holds. The second equality of (6.5) is equivalent 18 GANNA KUDRYAVTSEVA

to (f, 1)(e, m)=(e, 1)(f, n) which is equivalent to (ef, m)=(ef, n) which holds if and + only if m = n. The description of σ follows. It is now clear that ([m] , m) = max[(e, m)]σ.

(2) Part (1) implies that τR = τE(FIR(M))⋊M (see (5.3)). The claim that τR ∈ V (R) now follows from Proposition 5.2. Applying Proposition 4.2, it follows that there is a morphism

(6.6) ηR : FRR(M) → E(FIR(M)) ⋊ M.

of restriction monoids such that τR = ηRιFRR(M). We are coming to our main result.

Theorem 6.7. Let M be a monoid and R a set of admissible identities. Then ηR is an isomorphism of restriction monoids.

It is enough to prove that ηR is surjective (i.e., that E(FIR(M)) ⋊ M is M-generated via τR) and that there is an M-canonical morphism ΨR : E(FIR(M)) ⋊ M → FRR(M).

7. Proof of the main result

7.1. The map u 7→ Du from FI(X) to P (F R(X)). Let X be a set. We aim to construct a map from FI(X) to P (F R(X)). As a first step, we construct a map from the free involutive monoid Finv(X) to P (F R(X)). Recall that elements of Finv(X) are words over −1 X ∪ X . We provide a recursive construction as to how to associate u ∈ Finv(X) with a projection Du ∈ F R(X). If u = 1, we put Du = 1. Let now |u| = n ≥ 1 and assume that for words v with |v| < n the elements Dv are already defined. We then put: ∗ (Dvx) , if u = vx, x ∈ X, (7.1) Du = + −1 (xDv) , if u = vx , x ∈ X. ∗ + In particular, for x ∈ X, we have Dx = x and Dx−1 = x . ′ Remark 7.1. The map u 7→ Du is inspired by the map θ : FG(X) → P (M), where FG(X) is the free group over X and M a restriction monoid, from [14]. In fact, we use the same construction as in [14], but with different domain and range. The following is immediate from (7.1).

Lemma 7.2. Let u,v,w ∈ Finv(X). If Du = Dv then Duw = Dvw. ∗ ∗ ∗ Observe that, by (2.5), Dvxy = ((Dvx) y) = (Dvxy) . By induction, this and its dual equality, involving the operation +, lead to the following equalities that hold for all w ∈ X∗: ∗ + (7.2) Dvw =(Dvw) and Dvw−1 =(wDv) .

Recall [26] that the free inverse monoid FI(X) can be realized as the quotient of Finv(X) by the congruence ρ generated by pairs (xx−1x, x), x ∈ X∪X−1, and(xx−1yy−1,yy−1xx−1), x, y ∈ X ∪ X−1.

Lemma 7.3. Let u, v ∈ Finv(X). Then: −1 (1) Duxx−1xv = Duxv for all x ∈ X ∪ X ; EXPANSIONS OF MONOIDS 19

−1 (2) Duxx−1yy−1v = Duyy−1xx−1v for all x, y ∈ X ∪ X . Proof. Due to Lemma 7.2 we can assume that v = 1. (1) Let x ∈ X. Using (2.7), we calculate: ∗ + ∗ + ∗ ∗ Duxx−1x = ((x(Dux) ) x) =(Dux x) =(Dux) = Dux.

Similarly one shows that Dux−1xx−1 = Dux−1 . + ∗ (2) Let x, y ∈ X. Applying (2.7), we have Duxx−1 = Dux and Dux−1x = Dux . Thus: + + + + Duxx−1yy−1 = Dux y = Duy x = Duyy−1xx−1 ; + ∗ ∗ + Duxx−1y−1y = Dux y = Duy x = Duy−1yxx−1 ; ∗ ∗ ∗ ∗ Dux−1xy−1y = Dux y = Duy x = Duy−1yx−1x.

Corollary 7.4. Let u, v ∈ Finv(X). If u ρ v then Du = Dv.

Let u ∈ FI(X). Due to Corollary 7.4 there is a well-deﬁned element Du ∈ P (F R(X)) that is given by (7.1).

7.2. Some properties of elements Du. We now establish some properties of elements Du that will be needed in the sequel. Applying (2.7) and (7.2), it follows that for any e ∈ E(FI(X)) and u ∈ X∗, we have: ∗ + (7.3) Du∗e = Deu∗ = Deu and Du+e = Deu+ = Deu . ∗ Observe that Du = Du∗ and Du = Du+ for all u ∈ X . Applying Lemma 7.2 and (7.3), it follows that, for all u ∈ X∗ and e ∈ E(FI(X)) we have: ∗ + (7.4) Due = Deu and Du−1e = Deu .

Lemma 7.5. Let u ∈ FI(X). Then Du = Du∗ . Proof. We apply induction on |u|. The case u = 1 is trivial. Let n ≥ 1 and assume that in the case where |u| < n the claim is proved. Let u = vx, x ∈ X ∪ X−1. If x ∈ X we have: ∗ Du∗ = Dx−1v∗x =(Dx−1v∗ x) (by (7.1)) + ∗ =(Dv∗ x x) (by (7.4)) + ∗ =(Dvx x) (by the induction hypothesis) ∗ =(Dvx) = Dvx = Du. (applying (7.1)) The case where x ∈ X−1 is treated similarly.

Lemma 7.6. Let v ∈ FI(X) and e ∈ E(FI(X)). Then DvDe = Dve. Proof. Let |e| denote the minimal length of a word u over X ∪ X−1 such that e = u−1u in FI(X). We argue by induction on n = |e|. For n = 0 the claim clearly holds. Let n ≥ 1 and assume that if |e| < n, the claim holds. We prove the claim for |e| = n. Let e = u−1u. Suppose that u = wx where x ∈ X. Then, in view of Lemma 7.5 and (2.6), we have: ∗ ∗ ∗ DvD(wx)∗ = DvDwx = Dv(Dwx) =(Dwx) Dv =(DwxDv) . 20 GANNA KUDRYAVTSEVA

On the other hand, ∗ Dv(wx)∗ = Dvx−1w∗x =(Dvx−1w∗ x) (by (7.1)) ∗ =(Dvx−1 Dw∗ x) (by the induction hypothesis) + ∗ + ∗ = ((xDv) Dw∗ x) =(Dw∗ (xDv) x) (applying (7.1)) ∗ =(DwxDv) , (by (2.4) and Lemma 7.5) −1 so that DvD(wx)∗ = Dv(wx)∗ . The case where u = wx , x ∈ X, is treated similarly. Let ψ : F R(X) → FI(X) be the X-canonical projection map.

∗ Lemmae 7.7. Let u ∈ FI(X). Then ψ(Du)= u . Proof. We apply induction on |u|. If e|u| = 0, the statement is obvious. We assume that ∗ ∗ ∗ −1 −1 ∗ ψ(Dv)= v and let x ∈ X. If u = vx we have ψ(Du)=(v x) = x v vx =(vx) , and if u = vx−1 we have ψ(D )=(xv∗)+ = xv−1vx−1 =(vx−1)∗, as needed. e u e ∗ + ∗ It is useful to thinke of ψ(Du) as of (· , , , 1)-decomposition of u ∈ FI(X) over X. In particular, we have shown that such a decomposition always exists. e −1 ∗ + ∗ ∗ Example 7.8. For u = xy yz we have Du = (((yx ) y) )z) ∈ P (F R(X)) and thus ∗ ∗ + ∗ ∗ u = ψ(Du) = (((yx ) y) )z) ∈ E(FI(X)).

7.3. Thee induced map u 7→ Du from FI(M) to P (FR(M)). Recall that FI(M) is a quotient of FI[M] by the congruence γ (see Deﬁnition 4.1). Likewise, FR(M) is a quotient of F R⌊M⌋ by the congruence δ (see Deﬁnition 6.1). e Proposition 7.9. Let u, v ∈ FIe [M]. If u γ v then Du δ Dv.

Proof. It suﬃcies to assume that u and v aree γ-equivalente in one step, that is, via a relation [m][n] = [mn]([m][n])∗ or its inverse relation ([m][n])−1 = ([mn]([m][n])∗)−1. We consider two cases. e Case 1. Let u = p[m][n]q and v = p[mn]([m][n])∗q. We show that

(7.5) Du δ Dv and Du−1 δ Dv−1 .

We ﬁrst show that Du δ Dv. Applyinge (2.6), we have:e ∗ ∗ ∗ ∗ ∗ Dp[m][n] =(Dp⌊em⌋⌊n⌋) δ (Dp⌊mn⌋(⌊m⌋⌊n⌋) ) =(Dp⌊mn⌋) (⌊m⌋⌊n⌋) . On the other hand, applying Lemmae 7.6 and (7.3) and (7.2), we have: ∗ ∗ ∗ Dp[mn]([m][n])∗ = Dp[mn]D([m][n])∗ = Dp[mn](⌊m⌋⌊n⌋) =(Dp⌊mn⌋) (⌊m⌋⌊n⌋) , so that Dp[m][n] δ Dp[mn]([m][n])∗ and thus, by Lemma 7.2, Du δ Dv. −1 −1 −1 −1 −1 −1 We now showe that Du−1 δ Dv−1 . We have u = qe [n] [m] p and v = q−1([m][n])∗[mn]−1p−1 which is convenient to rewrite as v−1 = q−1[mn]−1([m][n])+p−1. e Similarly as above, one shows that each of the elements Dq−1[n]−1[m]−1 and Dq−1[mn]−1([m][n])+ + + is δ-related to (⌊m⌋⌊n⌋) (⌊mn⌋Dq−1 ) . It follows that Du−1 δ Dv−1 . e e EXPANSIONS OF MONOIDS 21

Case 2. Let u = p([m][n])−1q and v = p([mn]([m][n])∗)−1q. Then u−1 = q−1[m][n]p−1 and v−1 = q−1[mn]([m][n])∗p−1, and this case is reduced to the previous one.

Hence there is a well-deﬁned map FI(M) → P (FR(M)), given by [e]γe 7→ [De]δe. 7.4. Proof of Theorem 6.7: the case where R = ∅. We are now ready to complete the proof of Theorem 6.7 in the special case where R = ∅. We denote η∅ and τ∅ by η and τ, respectively. We aim to prove that η : FR(M) → E(FI(M)) ⋊ M is an isomorphism. Proposition 7.10. The morphism η is surjective. That is, E(FI(M)) ⋊ M is generated (as a restriction monoid) by the set τ(M)= {([m]+, m): m ∈ M}. Proof. Let (e, m) ∈ E(FI(M)) ⋊ M. We show that (e, m) ∈ hτ(M)i with the help of induction on the smallest length n of a word u over [M] ∪ [M]−1 such that the value of u∗ = u−1u in FI(M) is e. If n = 0, we have e = 1, and thus (e, m) = (1, 1) which is the nullary operation in hτ(M)i. Hence the basis of the induction is established. Assume that n ≥ 1 and that the claim is proved for e which can be written as e = u∗ where |u| ≤ n − 1. Let e = v∗ with |v| = n. If v = u[a], a ∈ M, applying (6.4) we have: (e, 1) = ((u[a])∗, 1)=(u∗[a]+, a)∗ = ((u∗, 1)([a]+, a))∗ ∈hτ(M)i. If v = u[a]−1, a ∈ M, we similarly have: (e, 1) = ((u[a]−1)∗, 1) = (([a]u∗)+, 1) = (([a]+, a)(u∗, 1))+ ∈hτ(M)i. Now, (e, m)=(e, 1)([m]+, m) ∈hτ(M)i for any m satisfying e ≤ [m]+. We deﬁne a map

(7.6) Ψ: E(FI(M)) ⋊ M → FR(M), Ψ(e, m)= De⌊m⌋. We verify that Ψ is a morphism of restriction monoids. Clearly, it respects the identity element. Veriﬁcations that Ψ respects the unary operations ∗ and + amount to showing ∗ + the equalities D(e[m])∗ =(De⌊m⌋) and De = (De⌊m⌋) , respectively, for any m ∈ M and any e ∈ E(FI(M)) satisfying e ≤ [m]+. They follow applying (7.1) and Lemma 7.5: ∗ + + + + (De⌊m⌋) = De[m] = D(e[m])∗ , (De⌊m⌋) =(De⌊m⌋ ) = De⌊m⌋ = De[m]+ = De. It remains to verify that Ψ preserves the multiplication. We have: Ψ((e, m)(f, n)) = Ψ(e([m]f)+, mn) (by (6.4))

= De([m]f)+ ⌊mn⌋ (by the deﬁnition of Ψ)

= DeD([m]f)+ ⌊mn⌋ (by Lemma 7.6) + −1 ∗ = DeD(f[m]−1)∗ ⌊mn⌋ (since ([m]f) =(f[m] ) )

= DeDf[m]−1 ⌊mn⌋ (by Lemma 7.5) + = De(⌊m⌋Df ) ⌊mn⌋. (by (7.1)) 22 GANNA KUDRYAVTSEVA

On the other hand, + + Ψ((e, m))Ψ((f, n)) = De⌊m⌋Df ⌊n⌋ = De(⌊m⌋Df ) (⌊m⌋⌊n⌋) ⌊mn⌋. + + + Therefore, it is enough to show that (⌊m⌋Df ) =(⌊m⌋Df ) (⌊m⌋⌊n⌋) . We calculate: + + (⌊m⌋Df ) (⌊m⌋⌊n⌋) = Df[m]−1 D([m][n])+ (by (7.3) with e = 1)

= Df[m]−1([m][n])+ (by Lemma 7.6) −1 + + −1 = Df[n]+[m]−1 (since [m] ([m][n]) = [n] [m] ) + + = Df[m]−1 =(⌊m⌋Df ) , (since f ≤ [n] and by (7.1)) as needed. Let m ∈ M. We show that Ψη⌊m⌋ = ⌊m⌋. This is equivalent to Ψ([m]+, m) = ⌊m⌋, ∗ which rewrites to D[m]+ ⌊m⌋ = ⌊m⌋. Because D[m]+ ⌊m⌋ ≤ ⌊m⌋ and (D[m]+ ⌊m⌋) = ∗ D[m]+[m] = D[m] = ⌊m⌋ , we are done. We have shown that η is an isomorphism. Recall from (6.3) that ψ∅ : FR(M) → FI(M) is the M-canonical morphism of restriction monoids. In the sequel, we denote ψ∅ just by ψ.

Corollary 7.11. The map e 7→ De, e ∈ E(FI(M)), establishes an isomorphism between the semilattices E(FI(M)) and P (FR(M)). The inverse isomorphism is the map ψ|P (FR(M)) : P (FR(M)) → E(FI(M)). In particular,

ψ(De)= e for all e ∈ E(FI(M)) and

Dψ(e) = e for all e ∈ P (FR(M)). Proof. We ﬁrst note that the semilattice E(FI(M)) is isomorphic to the semilattice P (E(FI(M)) ⋊ M) via the map e 7→ (e, 1) (see Subsection 2.3). Because Ψ is an iso- Ψ morphism and Ψ(e, 1) = De, it follows that the map e 7→ (e, 1) 7→ De is an isomorphism from E(FI(M)) onto P (FR(M)). Furthermore, by Lemma 7.7 and Proposition 7.9 we have ψ(De)= e for all e ∈ E(FI(M)). The statement follows.

7.5. Proof of Theorem 6.7: the case of an arbitrary R. We now proceed to the proof of Theorem 6.7 for the case where R is an arbitrary set of admissible identities.

Proposition 7.12. Let R be a set of admissible identities and u, v ∈ FI(M). If u γR v then Du δR Dv. Proof. If R = ∅, the statement is trivial. We thus assume that R =6 ∅. It suﬃces to consider the case where u and v are γR-equivalent in one step. In view of Proposition 3.10, we can assume that u in v are connected by a relation w · e = w · f, where w ∈ X∗ and e, f ∈ P (F R(X∗)), or by its inverse.

Case 1. Let ﬁrst u = p [w · e]ιFI(M),f q and v = p [w · f]ιFI(M),f q where f : X → M is a map and p, q ∈ FI(M). Since ιFI(M) = ψιFR(M) with ψ being a morphism, Lemma 3.2 implies that

(7.7) [e]ιFI(M),f = ψ([e]ιFR(M),f ). EXPANSIONS OF MONOIDS 23

We observe that

Dp[w·e] = Dp[w] [e] (by Deﬁnition 3.1) ιFI(M),f ιFI(M),f ιFI(M),f

= Dp[w] D[e] (by Lemma 7.6) ιFI(M),f ιFI(M),f

= D ˆ Dψ([e] ) (by Deﬁnition 3.1 and (7.7)) p[f(w)] ιFR(M),f ˆ ∗ =(Dp⌊f(w)⌋) [e]ιFR(M),f (by (7.1) and Corollary 7.11) ˆ ∗ =(Dp⌊f(w)⌋[e]ιFR(M),f ) (by (2.6)) ∗ =(Dp[w · e]ιFR(M),f ) . (by Deﬁnition 3.1) ∗ Likewise, Dp[w·f] = (Dp[w · f]ιFR ,f ) . Therefore, in view of Lemma 7.2, we have ιFI(M),f (M) Du δR Dv, as needed. Similarly, one shows that Du−1 δR Dv−1 . Case 2. Let u and v be connected by the relation (w · e)−1 = (w · f)−1. Then u−1 and v−1 are connected by the relation w · e = w · f, which reduces this case to the previous one.

Therefore, the map FI(M) → P (FR(M)), u 7→ Du, gives rise to the induced map FIR(M) → P (FRR(M)), also denoted by u 7→ Du. ♮ The quotient map γR : FI(M) → FIR(M), in turn, induces a quotient map of restric- ⋊ ⋊ ♮ tion monoids E(FI(M)) M → E(FIR(M)) M, given by (e, m) 7→ (γR(e), m). It follows that there is an analogue of Proposition 7.10 for any set R of admissible identities: E(FIR(M)) ⋊ M is M-generated via the map τR. Consequently, the morphism Ψ of (7.6) induces the morphism

(7.8) ΨR : E(FIR(M)) ⋊ M → FRR(M), ΨR(e, m)= De⌊m⌋,

that satisﬁes ΨRηR = idFRR(M) (in particular, it is an isomorphism). This completes the proof of Theorem 6.7. The following consequence of Theorem 6.7 extends Corollary 7.11.

Corollary 7.13. Let R be a set of admissible identities. The map e 7→ De, e ∈ E(FIR(M)), establishes an isomorphism between the semilattices E(FIR(M)) and P (FRR(M)). The inverse isomorphism is given by the map

ψR|P (FRR(M)) : P (FRR(M)) → E(FIR(M)). In particular, ψR(De)= e for all e ∈ E(FIR(M)) and

DψR(e) = e for all e ∈ P (FRR(M)). Corollary 7.14. Let R be a set of admissible identities and M be a monoid. If the word problem in FIR(M) is decidable, so is the word problem in FRR(M).

Proof. Applying (2.4) and (4.2), we can rewrite any element of FRR(M) in the form e⌊m⌋,

where e ∈ P (FRR(M)) and m ∈ M. Since e = DψR(e), we have e⌊m⌋ = DψR(e)⌊m⌋. The-

orem 6.7 implies that DψR(e)⌊m⌋ = DψR(f)⌊m⌋ if and only if ψR(e) = ψR(f) in FIR(M). 24 GANNA KUDRYAVTSEVA

If we know the deduction of ψR(e) = ψR(f) in FIR(M), proofs of Corollary 7.4, Propo-

sition 7.9 and Proposition 7.12 provide an algorithm how to pass from e = DψR(e) to f = DψR(f) in FRR(M). We conclude this section with an example demonstrating how the known structure of FIR(M) sheds light onto the structure of FRR(M). 2 3 Example 7.15. Let M = ha | a = a i and R = Rm. The structure of FIRm (M) in this example is easily understood: it is the quotient of the free [a]-generated inverse monoid by the relation [a]2 = [a]3. In particular, its idempotents are described by one-rooted Munn trees ([31]) over [a] with at most two edges. It is thus easy to check that ∗ + ∗ + 2 ∗ 2 + E(FIRm (M)) = {1, [a] , [a] , [a] [a] , ([a] ) , ([a] ) }. ⋊ By Theorem 6.7 ηRm : FRRm (M) → E(FIRm (M)) M is an isomorphism, thus one

gets a handle on the structure of FRRm (M). For example, because ∗ + ∗ ∗ ∗ + ηRm ((⌊a⌋ ⌊a⌋) )= ηRm ((⌊a⌋ ⌊a⌋) ) = ([a] [a] , 1), we have (⌊a⌋∗⌊a⌋)+ =(⌊a⌋∗⌊a⌋)∗. On the other hand, (7.9) ⌊a⌋∗⌊a⌋=( 6 ⌊a⌋∗⌊a⌋)+, because ∗ + ∗ + ∗ + ∗ + ∗ + ηRm (⌊a⌋ ⌊a⌋) = ([a] , a) ([a] , a) = ([a] [a] , a) =6 ([a] [a] , 1) = ηRm ((⌊a⌋ ⌊a⌋) ).

In contrast to (7.9), in FIRm (M) we have: (7.10) [a]∗[a] = ([a]∗[a])+. Indeed, ([a]∗[a])([a]∗[a])([a]∗[a]) = [a]∗[a]3 = [a]∗[a]2 = [a]−1[a]3 = [a]−1[a]2 = [a]∗[a], so that ([a]∗[a])−1 = [a]∗[a] by the uniqueness of the inverse element, which implies (7.10).

This means that the map ψRm in this example is not injective (cf. Proposition 8.2).

8. Special cases Let M be a monoid and R a set of admissible identities. 8.1. The case where M embeds into a group. For a monoid M, we put ⌈M⌉ = {⌈m⌉: m ∈ M}. The free group FG⌈M⌉ is M-generated via the map ιFG⌈M⌉ : M → FG⌈M⌉, m 7→ ⌈m⌉. We deﬁne FG(M) to be the quotient of FG⌈M⌉ by the congruence generated by the relations (8.1) ⌈1⌉ = 1;

(8.2) ⌈m⌉⌈n⌉ = ⌈mn⌉(⌈m⌉⌈n⌉)∗, m, n ∈ M. Since FG⌈M⌉ is a group, (⌈m⌉⌈n⌉)∗ = 1, so that (8.2) is equivalent to ⌈m⌉⌈n⌉ = ⌈mn⌉, m, n ∈ M. It follows that the M-canonical map ιFG(M) : M → FG(M) is a morphism. EXPANSIONS OF MONOIDS 25

In particular, ιFG(M) satisfies any admissible identity. Thus, FGR(M) = FG(M) for any set of admissible identities (where FGR(M) is defined along the lines of Definition 4.1). The group FG(M) is known as the free group over the monoid M [7] (also called the fundamental group of M [29]). Lemma 8.1. Let R be a set of admissible identities. The maximum group quotient of FIR(M) is isomorphic to FG(M).

Proof. The definitions imply that FG(M) is a quotient of FIR(M). Let π be the corre- sponding congruence on FIR(M). We assume that u, v ∈ FIR(M) are such that u π v and show that u σ v. Since π is generated by the relations [m][m]−1 = 1 and [m][n] = [mn], m, n ∈ M, it is enough to consider the following cases. Case 1. We assume that u = p[m][m]−1q and v = pq. Because [m][m]−1 = [m]+ ≤ 1, it follows that u ≤ v, so that u σ v. Case 2. We assume that u = p[m][n]q and v = p[mn]q. Because [m][n] ≤ [mn], it follows that u ≤ v, so that u σ v. We have proved that π ⊆ σ, so that π = σ, by the minimality of σ. Proposition 8.2. Let M be a monoid and R a set of admissible identities. The following statements are equivalent: (1) M embeds into a group. (2) The morphism ψR : FRR(M) → FIR(M) of (6.3) is injective. Proof. The definition of FG(M) implies that M embeds into a group if and only if M embeds into FG(M), that is, the map ιFG(M) is injective. (1) ⇒ (2) We assume that the map ιFG(M) is injective. By the proof of Theorem 6.7 + (see (7.8)), any element of FRR(M) can be written in the form De⌊m⌋ where e ≤ [m] . + + Let ψR(De⌊m⌋) = ψR(Df ⌊n⌋) where e ≤ [m] and f ≤ [n] . In view of Corollary 7.13 this yields e[m] = f[n]. Since e[m] σ f[n], we obtain ⌈m⌉ = ⌈n⌉, by Lemma 8.1. The assumption implies that m = n whence e[m]= f[m]. But then e =(e[m])+ =(f[m])+ = f. It follows that De⌊m⌋ = Df ⌊n⌋, so that the map ψR is injective. (2) ⇒ (1) We assume that the map ψR is injective. Let m, n ∈ M be such that ⌈m⌉ = ⌈n⌉. Then [m] σ [n] in FIR(M). By the definition of σ there are e, f ∈ E(FIR(M)) such that e[m] = f[n]. Without loss of generality, we assume that e ≤ [m]+ and f ≤ [n]+. Observe that e[m] = ψRΨR(e, m) and, similarly, f[n] = ψRΨR(f, n). Since ψR is injective and ΨR is bijective, it follows that (e, m)=(f, n) which yields m = n, as needed. We obtain the following result. Theorem 8.3. Let M be a monoid that embeds into a group and R a set of admissible identities. Then FRR(M) is isomorphic to the restriction submonoid of FIR(M) consisting of elements e[m] where e ∈ E(FIR(M)) and m ∈ M. This is precisely the [M]-generated restriction submonoid of FIR(M). As a monoid, FRR(M) is generated by E(FIR(M)) and [M].

We remark that under the assumption of Theorem 8.3, FRR(M) is a full restriction submonoid of FIR(M), that is, it contains all idempotents of FIR(M). 26 GANNA KUDRYAVTSEVA

∗ ∗ Example 8.4. Let M = A . As already noted, FRRm (A ) is isomorphic to F R(A) and ∗ FIRm (A ) is isomorphic to FI(A). Theorem 8.3 implies that F R(A) is isomorphic to the submonoid of FI(A) generated by E(FI(A)) and {[a]: a ∈ A}. We thus recover the result of Fountain, Gomes and Gould [14] on the structure of F R(A). Therefore, Theorem 6.7 extends the result of [14] from A∗ to any monoid M and from the set Rm to any set R of admissible identities. 8.2. The case where M is an inverse monoid. Let S be an inverse monoid. In this

section we provide models for FRRm (S) and FRRs (S). Due to Theorem 6.7, it is enough

to provide models for FIRm (S) and FIRs (S). Since, obviously, FIRm (S) ≃ S, we have the following statement. ⋊ Proposition 8.5. Let S be an inverse monoid. Then FRRm (S) ≃ E(S) S. In particular,

if G is a group then FRRm (G) is a group isomorphic to G. Remark 8.6. We note that E(S) ⋊ S can be endowed with the structure of an inverse monoid, by putting (e, s)−1 = ((es)∗,s−1). Let and denote the unary operations of the induced restriction monoid structure. Observe that, unless s−1s = 1, (e, s) =(e, s)−1(e, s)=((es)∗,s−1s) =6 ((es)∗, 1)=(e, s)∗ and, similarly, (e, s) =(6 e, s)+. It follows that, unless S is a group, the image of S under ⋊ + ⋊ the morphism τRm : S → E(S) S, s 7→ (s ,s), does not generate E(S) S as an inverse monoid. For example, the element (e, s)∗ = ((es)∗, 1) does not belong to this image.

We now turn to the inverse monoid FIRs (S). The following result characterizes strong premorphisms from S to inverse monoids.

Proposition 8.7. Let S, T be inverse monoids and f : S → T , s 7→ fs, a premorphism. The following statements are equivalent: (1) f is strong; (2) f satisﬁes the following conditions: −1 (i) for all s ∈ S: fs−1 = fs ; (ii) for all s, t ∈ S: if s ≤ t then fs ≤ ft. ∗ −1 Proof. (1) ⇒ (2) We assume that f strong. If e ∈ E(S) then fefe = fe2 fe = fefe fe = fe, so that fe ∈ E(T ). Let s ∈ S. Then:

∗ ∗ −1 −1 + −1 ∗ fs fs = fs fsfs−1s = fs fs fs = fs fs = fs ,

∗ ∗ so that fs ≤ fs . Using this, we obtain: ∗ ∗ fsfs−1 fs = fsfs∗ fs = fsfs = fs. −1 It follows that fs and fs−1 are mutually inverse, so that fs−1 = fs , by the uniqueness of the inverse element. We have shown condition (i). To show condition (ii), we assume that s ≤ t, that is, that s = ts∗. Then, in view of ∗ ∗ fs ≤ fs , we have:

∗ ∗ ∗ ∗ ∗ ∗ ∗ ftfs = fts fs∗ = fsfs = fsfs fs = fsfs = fs, EXPANSIONS OF MONOIDS 27

so that fs ≤ ft, as needed. (2) ⇒ (1) We assume that f satisﬁes conditions (i) and (ii) of part (2). Then, for any s, t ∈ S, we have:

∗ ∗ fsft = fsftft ≤ fstft = fstft−1 ft ≤ fstt−1 ft ≤ fsft. ∗ It follows that fsft = fstft , so that f is right strong. By symmetry, it is also left strong. Hence f is strong.

Proposition 8.7 combined with [27, Theorem 6.10, Theorem 6.17] leads to a presentation for Spr, the Lawson-Margolis-Steinberg generalized preﬁx expansion of S [27].

pr Theorem 8.8. For any inverse monoid S, S is isomorphic to FIRs (S). In particular, Spr is generated by the set [S]= {[s]: s ∈ S}, subject to the deﬁning relations: (1) [1] = 1; (2) [s][t] = [st][t]∗,[s][t] = [s]+[st], s, t ∈ S. Another presentation for Spr was obtained by Buss and Exel in [6]. Remark 8.9. It follows from the proof of Proposition 8.7 that the statement of Proposi- tion 8.7 (and thus of Theorem 8.8) remains valid in the semigroup setting, that is, in the absence of the identity element and the requirement that the identity element be preserved by a premorphism. From Theorem 6.7 and Theorem 8.8 we obtain the following. Corollary 8.10. Let S be an inverse monoid. Then: pr ⋊ FRRs (S) ≃ E(S ) S.

pr Using the known model for S [27, Proposition 6.16] this gives a model for FRRs (S).

There is an analogue of Remark 8.6 for FRRs (S): it can be similarly endowed with the structure of an inverse monoid, however, unless S is a group, τRs (S) does not generate ⋊ E(FIRs (S)) S as an inverse monoid.

Corollary 8.11. Let G be a group. Then FRRs (G) is an inverse monoid isomorphic to R each of FIRs (G) and G .

Proof. The isomorphisme FRRs (G) ≃ FIRs (G) follows from Theorem 8.3. The isomor- R pr R phism FIRs (G) ≃ G follows from Theorem 8.8 and G ≃ G . e e Acknowlegement The author is in debt to the referee for pointing out that admissible relations should be independent of a speciﬁc monoid M, which inspired me to introduce admissible identities. I am also grateful for the referee’s comments and suggestions, which led to a considerable improvement of the presentation of my ideas and results. 28 GANNA KUDRYAVTSEVA

References [1] F. Abadie, Enveloping actions and Takai duality for partial actions, J. Funct. Anal. 197 (1) (2003), 14–67. [2] J.-C. Birget, J. Rhodes, Almost finite expansions of arbitrary semigroups, J. Pure Appl. Algebra 32 (1984) (3), 239–287. [3] J.-C. Birget, J. Rhodes, Group theory via global semigroup theory, J. Algebra 120 (1989), 284–300. [4] M. J. J. Branco, G. M. S. Gomes, The generalized Szendrei expansion of an R-unipotent semigroup, Acta Math. Hungar. 123 (1-2) (2009), 11–26. [5] S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics 78, Springer, New York–Berlin, 1981. [6] A. Buss, R. Exel, Inverse semigroup expansions and their actions on C∗-algebras, Illinois J. Math. 56 (4) (2012), 1185–1212. [7] A. H. Clifford, G. B. Preston, The Algebraic Theory of Semigroups, Mathematical Surveys 7, Amer. Math. Soc., Providence, RI, Vol. 1 (1961), Vol. 2 (1967). [8] C. Cornock, Restriction Semigroups: Structure, Varieties and Presentations, Ph.D. Thesis, Uni- versity of York, 2011, available online at http://etheses.whiterose.ac.uk/1620/1/Thesis Claire Cornock Final.pdf [9] C. Cornock, V. Gould, Proper two-sided restriction semigroups and partial actions, J. Pure Appl. Algebra 216 (2012), 935–949. [10] M. Dokuchaev, Recent developments around partial actions, São Paulo J. Math. Sci. 13 (1) (2019), 195–247. [11] R. Exel, Partial actions of groups and actions of inverse semigroups, Proc. Amer. Math. Soc. 126 (1998), 3481–3494. [12] J. Fountain, G. M. S. Gomes, The Szendrei expansion of a semigroup, Mathematika 37 (2) (1990), 251–260. [13] J. Fountain, G. M. S. Gomes, V. Gould, Enlargements, semiabundancy and unipotent monoids, Comm. Algebra 27 (2) (1999), 595–614. [14] J. Fountain, G. M. S. Gomes, V. Gould, The free ample monoid, Internat. J. Algebra Comput. 19 (2009), 527–554. [15] G. M. S. Gomes, The generalized prefix expansion of a weakly left ample semigroup, Semigroup Forum 72 (3) (2006), 387–403. [16] G. M. S. Gomes, V. Gould, Graph expansions of unipotent monoids, Comm. Algebra 28 (1)(2000), 447–463. [17] V. Gould, Graph expansions of right cancellative monoids, Internat. J. Algebra Comput. 6 (6) (1996), 713–733. [18] V. Gould, Notes on restriction semigroups and related structures, available online at http://www- users.york.ac.uk/∼varg1/restriction.pdf [19] V. Gould, C. Hollings, Partial actions of inverse and weakly left E-ample semigroups, J. Aust. Math. Soc. 86 (3) (2009), 355–377. [20] C. Hollings, Partial actions of monoids, Semigroup Forum 75 (2) (2007), 293–316. [21] C. Hollings, Conditions for the prefix expansion of a monoid to be (weakly) left ample. Acta Sci. Math. (Szeged) 73 (3-4) (2007), 519–545. [22] C. Hollings, From right P P monoids to restriction semigroups: a survey, European J. Pure Appl. Math. 2 (2009), 21–57. [23] P. Jones, Almost perfect restriction semigroups, J. Algebra 445 (2016), 193–220. [24] J. Kellendonk, M. V. Lawson, Partial actions of groups, Internat. J. Algebra Comput. 14 (1) (2004), 87–114. [25] G. Kudryavtseva, Partial monoid actions and a class of restriction semigroups, J. Algebra, 429 (2015), 342–370. EXPANSIONS OF MONOIDS 29

[26] M. V. Lawson, Inverse semigroups. The theory of partial symmetries, World Scientiﬁc Publishing Co., Inc., River Edge, NJ, 1998. [27] M. V. Lawson, S. W. Margolis, B. Steinberg, Expansions of inverse semigroups, J. Aust. Math. Soc. 80 (2) (2006), 205–228. [28] S. W. Margolis, J. C. Meakin, E-unitary inverse monoids and the Cayley graph of a group presentation, J. Pure Appl. Algebra 58 (1) (1989), 45–76. [29] S. W. Margolis, J.-E. Pin, Inverse semigroups and extensions of groups by semilattices, J. Algebra 110 (2) (1987), 277–297. [30] D. Milan, B. Steinberg, On inverse semigroup C∗-algebras and crossed products, Groups Geom. Dyn. 8 (2) (2014), 485–512. [31] W. D. Munn, Free inverse semigroups, Proc. London Math. Soc. (3) 29 (1974), 385–404. [32] M. Petrich, N. R. Reilly, A representation of E-unitary inverse semigroups, Quart. J. Math. Oxford Ser. (2) 30 (119) (1979), 339–350. [33] M. B. Szendrei, A note on Birget-Rhodes expansion of groups, J. Pure Appl. Algebra, 58 (1989), 93–99. [34] M. B. Szendrei, Embedding into almost left factorizable restriction semigroups, Comm. Algebra 41 (4) (2013), 1458–1483. [35] M. B. Szendrei, Embedding of a restriction semigroup into a W -product, Semigroup Forum 89 (1) (2014), 280–291.

G. Kudryavtseva: University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova cesta 2, SI-1000 Ljubljana, Slovenia and Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia E-mail address: [email protected], [email protected]