On non-regular semigroups: $\lambda$-semidirect products, Zappa-Szép products and representations

This article encompasses the study of non-regular semigroups. First, we show that the $\lambda$-semidirect product $M \rtimes^{\lambda} W$ of a left $P$-Ehresmann semigroup $M$ and a left restriction semigroup $W$ is a left $P$-Ehresmann semigroup. We explore the behavior of generalized Green's relations on $M \rtimes^{\lambda} W$, and investigate some properties of $M \rtimes^{\lambda} W$. Second, the Zappa-Szép product of a right Ehresmann semigroup and its distinguished semilattice is studied. Lastly, the theory of representations of left Ehresmann semigroups with zero via homomorphisms of left Ehresmann semigroups with zero into Clifford restriction semigroups with zero is presented.