nLab inverse semigroup


This entry is about semigroups with a unary operator ii such that si(s)s=ss \cdot i(s) \cdot s = s and i(s)si(s)=i(s)i(s) \cdot s \cdot i(s) = i(s). For the semigroup with two-sided inverses, see instead at invertible semigroup.



An inverse semigroup is a semigroup SS (ie. a set equipped with an associative binary operation) such that for every element sSs\in S, there exists a uniqueinverses *Ss^*\in S such that ss *s=ss s^* s = s and s *ss *=s *s^* s s^* = s^*. It is evident from this that s **=ss^{\ast\ast} = s.


Needless to say, a group is an inverse semigroup. More to the point however:

  • The fundamental example is the following: for any set XX, let I(X)I(X) be the set of all partial bijections on XX, i.e. bijections between subsets of XX. The composite of partial bijections is their composite as relations (or as partial functions). In fact, any inverse semigroup is isomorphic to a sub-inverse-semigroup of I(X)I(X) by the “Cayley’s theorem” for inverse semigroups (see below).

This inverse semigroup plays a role in the theory similar to that of permutation groups in the theory of groups. It is also paradigmatic of the general philosophy that

Groups describe global symmetries, while inverse semigroups describe local symmetries.

Other examples include:

  • If XX is a topological space, let Γ(X)I(X)\Gamma(X)\subseteq I(X) consist of the homeomorphisms between open subsets of XX. Then Γ(X)\Gamma(X) is a pseudogroup of transformations on XX (a general pseudogroup of transformations is a sub-inverse-semigroup of Γ(X)\Gamma(X)).

  • If LL is a meet-semilattice, then LL is an inverse semigroup under the meet operation.


Lots to say here: the meet-semilattice of idempotents, the connection with ordered groupoids, various representation theorems.

Idempotents form a subsemigroup


For any xx in an inverse semigroup, xx *x x^\ast and x *xx^\ast x are idempotent. If ee is idempotent, then e *=ee^\ast = e.

The proof is trivial.


In an inverse semigroup, the product of any two idempotents e,fe, f is idempotent, and any two idempotents commute.


One easily checks that (ef) *=f(ef) *e(e f)^\ast = f(e f)^\ast e, and that f(ef) *ef(e f)^\ast e is an idempotent. So (ef) *(e f)^\ast is idempotent; as a result, (ef) *=ef(e f)^\ast = e f. Thus efe f and similarly fef e are idempotent. Next we have

ef(fe)ef=ef 2e 2f=efef=ef,fe(ef)fe=fe 2f 2e=fefe=fee f (f e) e f = e f^2 e^2 f = e f e f = e f, \qquad f e (e f) f e = f e^2 f^2 e = f e f e = f e

since e,f,ef,fee, f, e f, f e are all idempotent, and so fe=(ef) *=eff e = (e f)^\ast = e f, which completes the proof.

Thus the idempotents in an inverse semigroup form a subsemigroup which is commutative and idempotent. Such a structure is the same as a meet-semilattice except for the fact that there might not have an empty meet or top element; that is, we define an order \leq on idempotents by efe \leq f if and only if e=efe = e f, whence multiplication of idempotents becomes the binary meet.

*\ast is an anti-involution


For any two elements x,yx, y in an inverse semigroup, (xy) *=y *x *(x y)^\ast = y^\ast x^\ast.


Since the idempotents x *x,yy *x^\ast x, y y^\ast commute, we have

xy(y *x *)xy=x(yy *)(x *x)y=xx *xyy *y=xyx y (y^\ast x^\ast) x y = x (y y^\ast)(x^\ast x) y = x x^\ast x y y^\ast y = x y

and similarly y *x *(xy)y *x *=y *yy *x *xx *=y *x *y^\ast x^\ast (x y)y^\ast x^\ast = y^\ast y y^\ast x^\ast x x^\ast = y^\ast x^\ast, which is all we need.

Order structure


For elements x,yx, y in an inverse semigroup, the following are equivalent:

  1. There exists an idempotent ee such that x=eyx = e y,
  2. x=xx *yx = x x^\ast y,
  3. There exists an idempotent ff such that x=yfx = y f,
  4. x=yx *xx = y x^\ast x.

We show 3.2.3. \Rightarrow 2.; a similar proof shows 1.4.1. \Rightarrow 4. Clearly then we have \Rightarrow 2. \Rightarrow 1. \Rightarrow 4. \Rightarrow 3.

Given an idempotent ff such that x=yfx = y f, we have

x = yf = yy *yf = yfy *y sinceidempotentscommute = yffy *y = yff *y *y Proposition 1 = yf(yf) *y Lemma 2 = xx *y \array{ x & = & y f & \\ & = & y y^\ast y f & \\ & = & y f y^\ast y & since \; idempotents \; commute \\ & = & y f f y^\ast y & \\ & = & y f f^\ast y^\ast y & \text{Proposition 1} \\ & = & y f (y f)^\ast y & \text{Lemma 2} \\ & = & x x^\ast y & }

which gives 3.2.3. \Rightarrow 2.

A preorder \leq is defined on an inverse semigroup by saying xyx \leq y if any of the four conditions of Proposition is satisfied; transitivity follows by equivalence to 1. and closure of idempotents under multiplication. When restricted to idempotents, this preorder coincides with the meet-semilattice order.


If aba \leq b and xyx \leq y in an inverse semigroup, then axbya x \leq b y and x *y *x^\ast \leq y^\ast.


Writing a=eba = e b for some idempotent ee, we have ax=e(bx)a x = e (b x) and so axbxa x \leq b x. Similarly bxbyb x \leq b y, so axbya x \leq b y by transitivity. This gives axbya x \leq b y. If x=eyx = e y for an idempotent ee, then x *=(ey) *=y *e *x^\ast = (e y)^\ast = y^\ast e^\ast; this gives x *y *x^\ast \leq y^\ast,


The preorder \leq on an inverse semigroup is a partial order, i.e., if xyx \leq y and yxy \leq x, then x=yx = y.


From xyx \leq y we derive x *y *x^\ast \leq y^\ast and xx *yy *x x^\ast \leq y y^\ast, and similarly from yxy \leq x we derive yy *xx *y y^\ast \leq x x^\ast. Thus xx *=yy *x x^\ast = y y^\ast since the preorder on idempotents is a meet-semilattice, which is a partial order. Then from xyx \leq y we derive x=xx *y=yy *y=yx = x x^\ast y = y y^\ast y = y.

Thus an inverse semigroup is naturally regarded as an internal semigroup in the category of posets (equivalently, a finite-product preserving functor from the Lawvere theory of semigroups to Pos).

Relation to ordered groupoids

In this section, an ordered groupoid means an internal groupoid in the finitely complete category of posets Pos. For any finitely complete category CC, we observe that the forgetful functor Gpd(C)SemiCat(C)Gpd(C) \to SemiCat(C), taking an internal groupoid in CC to the underlying semicategory (remembering only composition of morphisms, forgetting presence of inverses and identity morphisms), has a right adjoint which takes a semicategory to the core groupoid of the category of idempotents attached to a semicategory (see here for details). (This observation is formulated in finite limit logic, and thus by a Yoneda lemma argument, its validity reduces to that of the observation in the special case C=SetC = Set.)

In particular, this construction may be applied to an inverse semigroup seen as a semigroup in PosPos:


The groupoid Ind(S)Ind(S) attached to an inverse semigroup SS is the core of the category of idempotents Idem(S)Idem(S) of SS, which as a semigroup in PosPos is viewed as a one-object semicategory BSB S in PosPos.

In more detail: an arrow eee \to e' in Idem(S)Idem(S) is a triple (e,x,e)(e, x, e') of elements in SS, where e,ee, e' are idempotent elements and xx is an element such that xe=x=exx e = x = e' x. Such an arrow is invertible precisely when e=x *xe = x^\ast x and e=xx *e' = x x^\ast, with inverse x *x^\ast. Thus the core consists of such arrows x:x *xxx *x: x^\ast x \to x x^\ast.

A key example to keep in mind is the inverse semigroup of partial bijections ϕ\phi on a set, where the arrows of the corresponding groupoid are actual invertible maps dom(ϕ)range(ϕ)\dom(\phi) \to range(\phi) between subsets. In general, the object part of the associated groupoid is not just a poset, but a poset with binary meets.

The reason for the notation Ind(S)Ind(S) is that this ordered groupoid is a so-called inductive groupoid, defined as follows:


An inductive groupoid is an internal groupoid GG in PosPos with the following additional properties:

  • The object part G 0G_0 admits binary meets;

  • Given x:efx: e \to f in G 1G_1 and eee' \leq e in G 0G_0, there exists a unique x:efx': e' \to f in G 1G_1 with xxx' \leq x, called the restriction [x| *e][x|_\ast e'].

  • Given x:efx: e \to f in G 1G_1 and fff \geq f' in G 0G_0, there exists a unique x:efx': e \to f' in G 1G_1 with xxx \geq x', called the corestriction [f *|x][f' _\ast| x].

In fact conditions 2. and 3. in this definition are equivalent. A morphism of inductive groupoids GGG \to G' is an internal functor from GG to GG' in PosPos.

For GG an inductive groupoid, a tensor product :G 1×G 1G 1\otimes: G_1 \times G_1 \to G_1 may be defined by the rule

xy=[x| *dom(x)cod(y)][dom(x)cod(y) *|y]x \otimes y = [x|_\ast dom(x) \wedge cod(y)] \cdot [dom(x) \wedge cod(y) _\ast|y]

where \cdot indicates composition in GG. It may be shown that (G 1,)(G_1, \otimes) is an inverse semigroup Inv(G)Inv(G), and the two notions are equivalent:


(Ehresmann-Schein-Nampooripad) There are canonical isomorphisms SInv(Ind(S))S \to Inv(Ind(S)) and GInd(Inv(G))G \to Ind(Inv(G)), providing an equivalence of categories InvSemiGrpIndGpdInvSemiGrp \simeq IndGpd.

Warning on the definition

With only a subtle change in definition, the result is that one gets only groups:


Let SS be an inhabited semigroup with the property that for every aSa \in S there exists a unique xSx \in S such that axa=aa x a = a. Then SS is a group.


Since SS is inhabited, say by an element bb, it has an idempotent ee, for example bb *b b^\ast. We will show that xe=xx e = x for any xx; by a similar argument ex=xe x = x, so that any idempotent ee is an identity (the identity 11), whence the idempotents aa *a a^\ast and a *aa^\ast a equal 11 for any aa and SS is a group.

If aya=aa y a = a for unique yy, then from (aya)ya=aya=a(a y a) y a = a y a = a it follows yay=yy a y = y and hence SS is an inverse semigroup. The same observation means it is enough to show (xe)x *(xe)=xe(x e) x^\ast (x e) = x e, since then also x *(xe)x *=x *x^\ast (x e) x^\ast = x^\ast, which by uniqueness implies xe=xx e = x.

The above results on inverse semigroups apply and we derive

xex *xe = xeex *xe = xee *x *xe Proposition1 = xe(xe) *xe Lemma2 = xe\array{ x e x^\ast x e & = & x e e x^\ast x e & \\ & = & x e e^\ast x^\ast x e & Proposition \; 1 \\ & = & x e (x e)^\ast x e & Lemma 2 \\ & = & x e }

as was to be shown.

Wagner-Preston Theorem: “Cayley’s Theorem” for inverse semigroups


Every inverse semigroup SS can be realized as a semigroup of partial bijections on a set.


First use the conventional Cayley’s theorem to embed SS in End(S)\text{End}(S) through the map sending sSs\in S to the map xsxx\mapsto sx. We can now consider SS a subset of End(S)\text{End}(S).

For each map sSs\in S, s *ss *=s *s^\ast s s^\ast=s^\ast, so s| s *(S):s *(S)s(S)s|_{s^\ast(S)}:s^\ast(S)\rightarrow s(S) has left inverse s *| s(S):s(S)s *(S)s^\ast|_{s(S)}:s(S)\rightarrow s^\ast(S). Similarly, as ss *s=ss s^\ast s=s, this map is a right inverse, so s| s *(S)s|_{s^\ast(S)} is a bijection.

Consider the map SI(S)S\rightarrow I(S) defined by s(s| s *(S):s *(S)s(S))s\mapsto (s|_{s^\ast(S)}: s^\ast(S)\rightarrow s(S)). We now check that this map is a homomorphism. For s,tSs,t\in S, we want to compose the partial bijections s| s *(S)s|_{s^\ast(S)} and t| t *(S)t|_{t^\ast(S)}. To do this, we first compute the overlap between the codomain of the former and the domain of the latter, which is s(S)t *(S)s(S)\cap t^\ast (S). The domain of t| t *(S)s| s *(S)t|_{t^\ast(S)}\circ s|_{s^\ast(S)} is then s *(s(S)t *(S))=s *t *(S)=(ts) *(S)s^\ast (s(S)\cap t^\ast (S))=s^\ast t^\ast(S)=(ts)^\ast (S) as required. Finally, we check injectivity. If s,tSs,t\in S and s s *(S)=t t *(S)s_{s^\ast(S)}=t_{t^\ast(S)}, then s *(S)=t *(S)s^\ast(S)=t^\ast(S) so ss *=ts *ss^\ast=ts^\ast. But then s *ss *=s *ts *s^\ast s s^\ast = s^\ast t s^\ast, so s=ts=t.


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The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis.

Last revised on April 27, 2024 at 11:46:22. See the history of this page for a list of all contributions to it.