Clifford semigroups

Clifford semigroups are completely regular, inverse, divisibility commutative semigroups. Clifford semigroups are useful in the study of divisibility commutative semigroups (semigroups for which L=R). They are a little known class of semigroups with a suprisingly elegant theory. For the purposes of this discussion, we will focus on Clifford semigroups of finite order.

Cycle notation

The elements of an inverse semigroup can be represented as charts, or partial symmetries. These are the binary relations that are one to one in the sense that both the inputs and the outputs have no repetitions among themselves. The composition operation then is the composition of binary relations which is defined so that if a relation does not have an element as an input then that element goes to nothing. The empty binary relation therefore is a zero element of relation composition, because anything composed with the empty relation is the empty relation.

In the general case, partial symmetries need not have the same inputs and outputs. When they have the same inputs and outputs then they are partial permutations and not simply partial symmetries. The common set of inputs and outputs is simply the domain of the partial permutation. If a partial symmetry has different inputs and outputs, then after a single composition some of the inputs will vanish and the element will never be able to return to itself under composition. Therefore, it clearly cannot be part of a completely regular semigroup. Clifford semigroups are the semigroups of partial permutations so defined.

This produces an issue of notation. Group theorists are typically used to simply omitting the domain of permutations of permutation groups. The domain is assumed to come from the ground set of the group, but this doesn't work with Clifford semigroups where permutations can carry around different domains then one another. There are two different ways that we can denote a partial permutation: we can give each partial permutation a set or we can include all the one-cycles which would otherwise be omitted in the cycle notations. Since associating a set to every single permutation is potentially cumbersone, the later notation is preferred. That means that the cycle (0 1) on the domain #{0 1 2 3} is represented as the partial permutation (0 1) (2) (3) and the partial permutation (0 1) is the cycle on the smaller domain of only two elements.

Limits of composition:

The partial permutations that are composed together must match up in their common domains. So (0 1) composed in a last-first order with (1 2) is the binary relation {(2 0)} which is not a partial permutation because it has a set of inputs of {2} and a set of outputs of {0} which are clearly not equal as sets. On the other hand, (2 3) (0 1) composed with (0 1) (4 5) matches up because it produces the identity permutation (0) (1). Basically, the intersection domain of the two composed permutations must be a union of a subset of the connected components of the first permutation and the same for the second permutation. Otherwise, the partial permutations can produce elements outside of their domain.

This means that there is no symmetric clifford semigroup. In order to get something similar to that, it is necessary to go to the broader context of the symmetric inverse semigroup which consists of all partial symmetries without the limitation that they need to be permutations. But the symmetric inverse semigroup is not divisibility commutative except in the trivial case. Clifford semigroups can embedded within the symmetric inverse semigroup by taking certain completely regular subsets, but the join of two Clifford semigroups in the subsemigroups lattice of a symmetric inverse semigroup need not be Clifford.

The algebraic preorder:

Every single Clifford semigroup is associated with a natural algebraic preorder. An element x is less then a greater element y if there exists some c such that cx=y or xc=y or there exists c,d such that cxd=y. In other words, an element is less then another one if it can be used to express the greater element in some form in an expression. This completely matches up with the familiar algebraic preordering on commutative semigroups and because of divisibility commutative this is the only algebraic preorder that needs to be associated with the semigroup. Divisibility commutative allows us to treat certain non-commutative semigroups similarly to commutative ones.

Semilattices:
When we make the algebraic preorder antisymmetric we arrive semilattices. In a semilattice every element is idempotent, which means that as a partial permutation every element is going to be an identity permutation. The semilattice on the permutations {(), (0), (4), (5), (4)(5), (0)(1), (0)(2), (0)(3),(0)(1)(2)} is displayed below.



In these cases, the semigroup operation is simply the intersection of the domains of the identity permutations. So these are essentially intersection semilattices except that elements are represented as identity permutations rather then as sets.

Groups:
Clifford semigroups in which every partial permutation has the same domain are naturally permutation groups. The only difference is that the partial permutations include all their one cycles so that the full domain is specified. So for example, the Clifford semigroup {(0)(1)(2)(3),(0 1)(2)(3),(0)(1)(2 3),(0 1)(2 3)} forms a group because every partial permutation has the same domain. In that case, optionally we can ommit the single cycles and use the simpler group cycle notation. In a larger Clifford semigroup, it is easy to see that every identity permutation forms a subgroup around it. These maximal subgroups are clearly the H classes of the identity permutations, which are ensured to always form groups.

Examples

These exmaples will use the traditional table notation to denote semigroups. These semigroups can then be related to partial permutations that could generate them. The below semigroup is equivalent to the Clifford semigroup {(1)(2),(1 2),(1)(2)(3)(4),(1)(2)(3 4)}.

[ [ 1, 2, 1, 1 ], 
  [ 2, 1, 2, 2 ], 
  [ 1, 2, 3, 4 ], 
  [ 1, 2, 4, 3 ] ]

That semigroup is equally to the two cyclic groups ordinally summed together. The below semigroup also has two cyclic groups chained together in order, but they are chained together differently so it produces a semigroup. This can be seen from its partial permutations representation: {(1)(2),(1 2),(1)(2)(3)(4),(1 2)(3 4)}. The only difference is the partial permutation (1 2)(3 4) now has action on the cyclic subgroup above it.

[ [ 1, 2, 1, 2 ], 
  [ 2, 1, 2, 1 ], 
  [ 1, 2, 3, 4 ], 
  [ 2, 1, 4, 3 ] ]
 

Groups can always be ordinally combined together in any order to get a semilattice of groups. The reason that Clifford semigroups are non-trivial is that they can be combined together in different ways as demonstrated above in the simplest case. The action of a lower group on an upper group chained together in order is always determined by the composition of the lower element with the upper identity, which produces the permutation representation in the upper group. So lower elements can subsume some of the role of upper elements.