This text carries out some ideas about EXACT and d P− EXACT sequences of transformation semi groups. Some theorems like the short five lemma (Lemma 1.3 and Lemma 2.3) are valid here as in EXACT sequences of R − modules.

Let λ be an infinite regular cardinal and A a locally λ-presentable additive category. We show that any λ-pure morphism (resp. λ-pure quotient) in A creates a kernel-cokernel pair. This implies that the class of all λ-pure kernel-cokernel pairs in A forms an EXACT structure. Additionally, we will describe λ-pure kernel-cokernel pairs in A and will prove that any λ-directed diagram of objects in A induces a canonical λ-pure kernel-cokernel pair.

In this article, we show the existence of certain EXACT sequences with respect to two homology theories, called d-homology and extended d-homology. We present su cient conditions for the existence of long EXACT extended d-homology sequence. Also we give some illustrative examples.

We examine the diffusion equation on the sphere. In this sense, we answer the question of the symmetry classification. We provide the algebra of symmetry and build the optimal system of Lie subalgebras. We prove for one-dimensional optimal systems of Eq. (1. 4), space is expanding Ricci solitons. Reductions of similarities related to subalgebras are classified, and some EXACT invariant solutions of the diffusion equation on the sphere are presented.

The notion of quasi-EXACT sequence of modules was introduced by B. Davvaz and coauthors in 1999 as a generalization of the notion of EXACT sequence. In this paper we investigate further this notion. In particular, some interesting results concerning this concept and torsion functor are given.