Cross-connections in Clifford Semigroups

Grillet introduced cross-connections as a pair of functions to describe the interrelationship between the posets of principal left and right ideals of a regular semigroup. This construction involved building two intermediary semigroups and further identifying a fundamental image of the semigroup as a subdirect product, using the cross-connection functions. But isomorphic posets give rise to isomorphic cross-connections. So, Nambooripad replaced posets with certain small categories to overcome this limitation. Hence, using the categorical theory of cross-connections, Nambooripad constructed arbitrary semigroups from their ideal structure.

Starting from a regular semigroup $S$, Nambooripad identified two small categories: $\mathbb{L}(S)$ and $\mathbb{R}(S)$ which abstract the principal left and right ideal structures of the semigroup, respectively. He showed that these categories are interconnected using a pair of functors. It can be seen that the object maps of these functors coincides with Grillet’s cross-connection functions and so Nambooripad called his functors also cross-connections. Further, Nambooripad showed that this correspondence can be extended to an explicit category equivalence between the category of regular semigroups and the category of cross-connections. Hence the ideal structure of a regular semigroup can be completely captured using these ‘cross-connected’ categories.

Being a rather technical construction, it is instructive to work out the simplifications that arise in various special classes of semigroups. There has been several works in this direction [4, 2, 10, 3, 5, 1] and in this short article, we propose to outline how the construction simplifies in a couple of very natural classes of regular semigroups: namely Clifford semigroups and semilattices. As the reader may see, this exercise also provides some useful illustrations to several cross-connection related subtleties.

In fact, Clifford semigroups are one of the first classes of inverse semigroups whose structure was studied. It was originally defined [6] as a union of groups in which idempotents commute. It may be noted that some authors refer to general union of groups also as Clifford semigroups but we shall follow [8] and refer to a semilattice of groups as a Clifford semigroup. The following characterizations of Clifford semigroups will be useful in the sequel.

In Nambooripad’s cross-connection description, starting from a regular semigroup $S$, two small categories of principal left and right ideals (denoted by $\mathbb{L}(S)$ and $\mathbb{R}(S)$, respectively in the sequel) are defined. Then their inter-relationship is abstracted as a pair of functors called cross-connections. Conversely, given an abstractly defined pair of cross-connected categories (with some special properties), one can construct a regular semigroup. This correspondence between regular semigroups and cross-connections is proved to be a category equivalence.

The construction of the regular semigroup from the category happens in several layers and the crucial object here is the intermediary regular semigroup $T\mathbb{L}(S)$ of normal cones from the given category $\mathbb{L}(S)$. We shall see that when the semigroup $S$ is Clifford, then the semigroup $T\mathbb{L}(S)$ is isomorphic to $S$. It is known that this is not true in general [5] even for an inverse semigroup $S$. This in turn, highly degenerates the cross-connection structure in Clifford semigroups. This is discussed in the next section. In the last section, we specialise our discussion to an arbitrary semilattice and describe the cross-connections therein.

As mentioned above, since the article can be seen as a part of a continuing project of studying the various classes of regular semigroups within the cross-connection framework, we refer the reader to [5, 10, 1] for the preliminary notions and formal definitions in Nambooripad’s cross-connection theory. We also refer the reader to [9] for the original treatise on cross-connections.

Recall from [9] that given a regular semigroup $S$, the normal category $\mathbb{L}(S)$ of principal left ideals are defined as follows. The object set

and the morphisms in $\mathbb{L}(S)$ are partial right translations. In fact, the set of all morphisms between two objects $Se$ and $Sf$ may be characterised as the set $\{\rho(e,u,f):u\in eSf\}$, where the map $\rho(e,u,f)$ sends $x\in Se$ to $xu\in Sf$.

First, we proceed to discuss some special properties of the category $\mathbb{L}(S)$ when $S$ is a Clifford semigroup. This will lead us to the characterisation of the semigroup $T\mathbb{L}(S)$ of all normal cones in $S$.

Given the normal category $\mathbb{L}(S)$ of principal left ideals of a regular semigroup $S$, it is known that two morphisms are equal, i.e. $\rho(e,u,f)=\rho(g,v,h)$ if and only if $e\mathrel{\mathscr{L}}g$, $f\mathrel{\mathscr{L}}h$, and $v=gu$. Also, given two morphisms $\rho(e,u,f)$ and $\rho(g,v,h)$, they are composable if and only $Sf=Sg$ so that $\rho(e,u,f)\rho(g,v,h)=\rho(e,uv,h)$. Now, we see that the equality of morphisms simplify when $S$ is a Clifford semigroup.

Now, we proceed to characterise the building blocks of the cross-connection construction, namely the normal cones in the category $\mathbb{L}(S)$. An ‘order-respecting’ collection of morphisms in a normal category is defined as a normal cone.

Now, for each $a\in S$, we can define a function $\rho^{a}\colon v\mathbb{L}(S)\to\mathbb{L}(S)$ as follows:

It is easy to verify that the map $\rho^{a}$ is a well-defined normal cone with apex $Sf$ in the sense of Definition 2.1, see [9, Lemma III.15].

In the sequel, the normal cone $\rho^{a}$ is called the principal cone determined by the element $a$. In particular, observe that, for an idempotent $e\in E(S)$, we have a principal cone $\rho^{e}$ such that $\rho^{e}(Se)=\rho(e,e,e)=1_{Se}$. This leads us to the most crucial proposition of this article.

In general, for an arbitrary regular semigroup $S$, two distinct principal cones $\rho^{a}$ and $\rho^{b}$ may be equal in $\mathbb{L}(S)$ even when $a\neq b$. But when $S$ is Clifford, we proceed to show that it is not the case.

Recall from [9, Section III.1] that the set of all normal cones in a normal category forms a regular semigroup, under a natural binary operation. So, in particular, given the normal category $\mathbb{L}(S)$, the set $T\mathbb{L}(S)$ of all normal cones in the category $\mathbb{L}(S)$ is a regular semigroup. Now, we proceed to characterise this semigroup when $S$ is Clifford.

The above theorem characterises the semigroup of $T\mathbb{L}(S)$ of all normal cones in $\mathbb{L}(S)$; this naturally leads us to the complete description of the cross-connection structure of the semigroup $S$ as follows.

Recall from [9, Section III.4] that given a normal category, it has an associated dual category whose objects are certain set-valued functors and morphisms are natural transformations. Now we proceed to characterise the normal dual $N^{*}\mathbb{L}(S)$ of the normal category $\mathbb{L}(S)$ of principal left ideals of a Clifford semigroup $S$.

Dually, we can easily prove the following results:

Recall from [9, Theorem IV.1] that the cross-connection of a regular semigroup $S$ is defined as a quadruplet $(\mathbb{L}(S),\mathbb{R}(S),\Gamma,\Delta)$ such that $\Gamma\colon\mathbb{R}(S)\to N^{*}\mathbb{L}(S)$ and $\Delta\colon\mathbb{L}(S)\to N^{*}\mathbb{R}(S)$ are functors satisfying certain properties. Now, using Theorems 2.6 and 2.7, we see that both the functors $\Gamma$ and $\Delta$ are in fact isomorphisms. And hence the cross-connection structure degenerates to isomorphisms of the associated normal categories in a Clifford semigroup $S$.

Clearly, a semilattice is a Clifford semigroup. So, we specialise our discussion to a semilattice using the results in the previous section. The following theorem follows from Theorem 2.5.

Theorems 2.6 and 2.7 when applied to a semilattice can be unified as follows:

Hence, we see that both the cross-connections functors $\Gamma$ and $\Delta$ are isomorphisms, when we have a semilattice also. So, the cross-connection structure degenerates to isomorphisms of the associated normal categories in a semilattice, too.