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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.

The normal categories associated with unit regular semigroups and various classes of strongly unit regular semigroups were described as ${𝒰}{ℛ}$-categories with appropriate isomorphism extension properties. The normal categories associated with unit regular semigroups have been characterized as weak ${𝒰}{ℛ}$-categories. The cases of ${ℛ}$-strongly unit regular, ${ℒ}$-strongly unit regular and strongly unit regular semigroups were characterized using corresponding forms of isomorphism extensions. It is shown that when a normal category satisfies one of these isomorphism extension properties, then its normal dual satisfies a corresponding isomorphism extension property. Further, we give a description of the extension isomorphisms in the dual $N^{*}{𝒞}$ in terms of extensions in ${𝒞}$.

The concept of normal category was introduced by Nambooripad in his theory of cross connections for describing the structure of regular semigroups. These are essentially the categories of principal left ideals and the categories of principal right ideals of regular semigroups with appropriate morphisms. These categories also have an associated partial order on the set of objects. Discrete normal categories are normal categories in which the partial order on the object set is the identity relation. A cross connection relates two normal categories ${𝒞}$ and ${𝒟}$ and produces a regular semigroup $S$ such that one is the category of principal left ideals and the other the category of principal right ideals of the semigroup $S$. This paper considers the question of compatibility of two normal categories in order that they can be connected by a cross connection. It is known that given any normal category ${𝒞}$ there is a normal category ${𝒟}={{𝒩}}^{\ast }{𝒞}$ called the normal dual of ${𝒞}$ such that ${𝒞}$ is compatible with ${𝒟}$. The question of finding all normal categories ${𝒟}$ that are compatible with a given normal category ${𝒞}$ remains unsettled. Such categories ${𝒟}$ are called compatible duals of ${𝒞}$. For the class of discrete normal categories the characterization of all compatible duals has been done here.

In this paper, we introduce the normal category $\mathcal{ℒ}({𝒮})$ [$\mathcal{ℛ}({𝒮})$] of principal left [right] ideals of the normed algebra ${𝒮}$ of all finite rank bounded operators on a Hilbert space $H$ and show that $\mathcal{ℒ}({𝒮})$ and $\mathcal{ℛ}({𝒮})$ are isomorphic using Hilbert space duality. We also prove that the semigroup of all normal cones in $\mathcal{ℒ}({𝒮})$ is isomorphic to the semigroup of all finite rank operators on $H$. Further, we construct bounded normal cones in $\mathcal{ℒ}({𝒮})$ such that the set of all bounded normal cones in $\mathcal{ℒ}({𝒮})$ forms a normed algebra isomorphic to the normed algebra ${𝒮}$.

The principal left ideals of a regular semigroup have been abstractly described by Nambooripad as normal categories. A semigroup $S$ is said to be a band if $a^2=a$ for all $a\in S$. Since bands are regular semigroups, the category of principal left ideals is a normal category. This paper characterizes the normal categories which arise as categories of principal left ideals of bands.