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Benjamin Steinberg
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There is an extension due to Schutzenberger. One needs to interpret Lagrange correctly. A partial function on a set $X$ is a function defined on a subset of $X$ to $X$. The composition of partial functions is defined where it makes sense.

The set of all partial functions on X is a semigroup and so we can define the notion of an action of a semigroup S$S$ by partial functions on the right of X. Say the action is transitive if for all $x\neq y$ in X there is s$s$ in S with xs=y$xs=y$.

Define an endomorphism of this action to be a totally defined map $f:X\to X$ such that $f(xs)=f(x)s$ (where equality means both sides are either undefined or agree).

Theorem.Theorem. If X$X$ is finite and $T$ is a semigroup of endomorphisms of the action of S$S$ on X$X$, then the size of X$X$ is divisible by the size of T$T$.

The proof is trivial.: Show T$T$ is a group acting freely on X$X$.

This generalizes Lagrange by taking G=X$S=G=X$ with the regular right action and T$T=H$ a subgroup acting on the left.

Schutzenberger used this to generalize the monomial representation of groups to semigroups.

There is an extension due to Schutzenberger. One needs to interpret Lagrange correctly. A partial function on a set $X$ is a function defined on a subset of $X$. The composition of partial functions is defined where it makes sense.

The set of all partial functions on X is a semigroup and so we can define the notion of an action of a semigroup S by partial functions on the right of X. Say the action is transitive if for all $x\neq y$ there is s in S with xs=y.

Define an endomorphism of this action to be a totally defined map $f:X\to X$ such that $f(xs)=f(x)s$ (where equality means both sides are either undefined or agree).

Theorem. If X is finite and is a semigroup of endomorphisms of the action of S on X, then the size of X is divisible by the size of T.

The proof is trivial. Show T is a group acting freely on X.

This generalizes Lagrange by taking G=X with the regular right action and T a subgroup acting on the left.

Schutzenberger used this to generalize the monomial representation of groups to semigroups.

There is an extension due to Schutzenberger. One needs to interpret Lagrange correctly. A partial function on a set $X$ is a function defined on a subset of $X$ to $X$. The composition of partial functions is defined where it makes sense.

The set of all partial functions on X is a semigroup and so we can define the notion of an action of a semigroup $S$ by partial functions on the right of X. Say the action is transitive if for all $x\neq y$ in X there is $s$ in S with $xs=y$.

Define an endomorphism of this action to be a totally defined map $f:X\to X$ such that $f(xs)=f(x)s$ (where equality means both sides are either undefined or agree).

Theorem. If $X$ is finite and $T$ is a semigroup of endomorphisms of the action of $S$ on $X$, then the size of $X$ is divisible by the size of $T$.

The proof is trivial: Show $T$ is a group acting freely on $X$.

This generalizes Lagrange by taking $S=G=X$ with the regular right action and $T=H$ a subgroup acting on the left.

Schutzenberger used this to generalize the monomial representation of groups to semigroups.

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Benjamin Steinberg
  • 38.6k
  • 3
  • 104
  • 186

There is an extension due to Schutzenberger. One needs to interpret Lagrange correctly. A partial function on a set $X$ is a function defined on a subset of $X$. The composition of partial functions is defined where it makes sense.

The set of all partial functions on X is a semigroup and so we can define the notion of an action of a semigroup S by partial functions on the right of X. Say the action is transitive if for all $x\neq y$ there is s in S with xs=y.

Define an endomorphism of this action to be a totally defined map $f:X\to X$ such that $f(xs)=f(x)s$ (where equality means both sides are either undefined or agree).

Theorem. If X is finite and is a semigroup of endomorphisms of the action of S on X, then the size of X is divisible by the size of T.

The proof is trivial. Show T is a group acting freely on X.

This generalizes Lagrange by taking G=X with the regular right action and T a subgroup acting on the left.

Schutzenberger used this to generalize the monomial representation of groups to semigroups.