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Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalized Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalization of the cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.

Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup $\mathrm{\text{Sing}}(X)$ of all non-invertible transformations on a set $X$. The categories involved are characterized as the powerset category $\mathcal{𝒫}(X)$ and the category of partitions $\mathrm{\Pi }(X)$. We describe these categories and show how a permutation on $X$ gives rise to a cross-connection. Further, we prove that every cross-connection between them is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to $\mathrm{\text{Sing}}(X)$. We also describe the right reductive subsemigroups of $\mathrm{\text{Sing}}(X)$ with the category of principal left ideals isomorphic to $\mathcal{𝒫}(X)$. This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of $\mathrm{\text{Sing}}(X)$.

A completely simple semigroup $S$ is a semigroup without zero which has no proper ideals and contains a primitive idempotent. It is known that $S$ is a regular semigroup and any completely simple semigroup is isomorphic to the Rees matrix semigroup $\mathcal{ℳ}[G;I,\mathrm{\Lambda };P]$ (cf. D. Rees, On semigroups, Proc. Cambridge Philos. Soc.36 (1940) 387–400). In the study of structure theory of regular semigroups, Nambooripad introduced the concept of normal categories to construct the semigroup from its principal left (right) ideals using cross-connections. A normal category ${𝒞}$ is a small category with subobjects wherein each object of the category has an associated idempotent normal cone and each morphism admits a normal factorization. A cross-connection between two normal categories ${𝒞}$ and ${𝒟}$ is a local isomorphism$\mathrm{\Gamma }:{𝒟}\to N^{\ast }{𝒞}$ where $N^{\ast }{𝒞}$ is the normal dual of the category ${𝒞}$. In this paper, we identify the normal categories associated with a completely simple semigroup $S=\mathcal{ℳ}[G;I,\mathrm{\Lambda };P]$ and show that the semigroup of normal cones $T{ℒ}(S)$ is isomorphic to a semi-direct product $G^{\mathrm{\Lambda }}⋉\mathrm{\Lambda }$. We characterize the cross-connections in this case and show that each sandwich matrix $P$ correspond to a cross-connection. Further we use each of these cross-connections to give a representation of the completely simple semigroup as a cross-connection semigroup.