Tangent-like Spaces to Local Monoids

The main new notions are the notions of tangent-like spaces and local monoids. The main result is the passage from a local monoid to its tangent-like space which is a local Leibniz algebra. Based on my belief that Leibniz algebras are too general to establish a fair counterpart of Lie theory in the context of Leibniz algebras, I introduced the notion of a local Leibniz algebra in Section 1.6 of [3]. The purpose of this paper is to construct the analogue of the passage from a linear Lie group to its Lie algebra in the context of local Leibniz algebras. The group-like objects I need in constructing the analogue are local monoids, which are obtained by adding more algebraic structures to monoids with diconjugations introduced in Section 4.2 of [2]. One of the difficulties I experienced in constructing the analogue is to find a suitable definition of a tangent space. The notion of a tangent-like space introduced in this paper is good enough for the purpose of this paper, but some changes may be needed in order to use it to develop the counterpart of differential geometry in a more general context. The paper consists of three sections. Section 1 discusses trimonoids, which give the algebraic foundation of this paper. Section 2 introduces the notion of a local monoid. Section 3 constructs the passage from a local monoid to its tangent-like space which is a local Leibniz algebra. Thoughout this paper, we will use Chapter 3 and Chapter 4 of [2].