A representation of a finite semigroup. Unlike the Semigroup typeclass, this does not refer to a type whose objects collectively form a semigroup. Instead, it is a type whose objects individually represent semigroups.

A semigroup presented as a multiplication table. Use getTable to extract the actual table; this type is otherwise opaque.

A semigroup presented as a collection of actions. The bases are the given set of generating actions, while the completion is the set of other semigroup elements.

Construct a semigroup from a set of generating transformations. Transformations are given as functions over an initial segment of the nonnegative integers. The transformation [a,b,c] maps 0 to a, 1 to b, and 2 to c.

Returns the set of element indices in the given transformation that map the given object into the given set.

Find a small set of basis elements such that none can be written in terms of the others and the entire semigroup can be written in terms of these bases.

The subsemigroup of the given semigroup generated by the indicated elements.

Create an action semigroup whose bases correspond to the given elements; for each element \(i\), there is a basis representing the transformation \(f(x)=xi\).

Return the local subsemigroup of the given semigroup \(S\) formed by wrapping elements with the given element \(i\). That is, all and only elements of the form \(isi\) for \(s\in S\) are included.

Return the local subsemigroups of the given semigroup which happen to be monoids. These are the ones whose wrapping element is idempotent. If the semigroup itself is a monoid, it is in the list.

Return a list of subsemigroups of the form \(e\cdot M_e\cdot e\) for idempotent \(e\), where \(M_e\) is the set of elements generated by those elements greater than or equal to \(e\) with respect to the \(\mathcal{J}\)-order.

Reverse the multiplication of the semigroup.

If the given semigroup is already a monoid, returns it. Otherwise, returns a monoid by adjoining a neutral element.

If the given semigroup is not a monoid, then it is returned as-is. Otherwise, the neutral element is removed if possible, that is, if it cannot be written in terms of other elements.

Given an element \(x\), return the set of elements representable as \(sxt\) for some \(s\) and \(t\) in \(S^1\), as a distinct ascending list.

The \(\mathcal{J}\)-order: \(x\leq y\) if and only if \(x=syt\) for some \(s\) and \(t\) in \(S^1\).

If the given semigroup is a monoid, return Just its neutral element. Otherwise return Nothing.

Insert a new element \(e\) such that for all elements \(x\), it holds that \(ex=x=xe\).

If the given semigroup has an element which is both a left- and right-zero, that is, an element e where \(ex=e=xe\) for all \(x\), return Just that zero. Otherwise, return Nothing.

Insert a new element \(e\) such that for all elements \(x\), it holds that \(ex=e=xe\).

Return the elements that square to themselves as an ascending list.

Return the unique idempotent power of the given element. The result of omega s i is the ith element of omegas s, if i is a valid element index.

Return an action mapping semigroup elements to their unique idempotent powers.

Partition the elements of the semigroup \(S\) such that distinct elements \(x\) and \(y\) are in the same region if and only if there are \(s\) and \(t\) such that \(y=xs\) and \(x=yt\).

Partition the elements of the semigroup \(S\) such that distinct elements \(x\) and \(y\) are in the same region if and only if there are \(s\) and \(t\) such that \(y=sx\) and \(x=ty\).