inverse semigroup


An inverse semigroup is a semigroup SS (a set with an associative binary operation) such that for every element sSs\in S, there exists a unique “inverse” s *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).

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 groupoid?s, various representation theorems.


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. We then 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, completing 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.


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.


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 partial order \leq is defined on an inverse semigroup by saying xyx \leq y if any of the four conditions of Proposition 2 is satisfied. When restricted to idempotents, this order 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.


We observe from Proposition 2 that for any elements c,xc, x we have cx *x=xx *cc x^\ast x = x x^\ast c. From the hypotheses a=aa *ba = a a^\ast b and x=yx *xx = y x^\ast x we have ax=aa *byx *x=(xx *)aa *bya x = a a^\ast b y x^\ast x = (x x^\ast) a a^\ast b y by our observation. But e=(xx *)(aa *)e = (x x^\ast)(a a^\ast) is idempotent by Lemma 1. This gives axbya x \leq b y. If x=eyx = e y for an idempotent ee, then x *=(ey) *=y *e *=y *ex^\ast = (e y)^\ast = y^\ast e^\ast = y^\ast e; this gives x *y *x^\ast \leq y^\ast,


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Revised on November 28, 2015 21:09:50 by Todd Trimble (