The Category Of L-Semilattices

We get a L-semilattice when a lattice acts on a semilattice, L-semilattice has its own particular structure and properties, and is widely used in logic, graph the-ory,combinatorial theory,computer science,automatic control,representation theory of algebras,operator algebra. The foundation theory of L-semilattice partly come from the S-systems theory of semigroups, partly come from the mod-ules theory of rings.We get a S-system when a semigroup acts on a set.Different semigroup will generate different S-system. In the S-systems theory of semigroup,specially the S-system theory of pomonoid, it is an important research method that study-ing the internal characteristics of semigroup from the external environment of the semigroup such as congruence lattice, the category of S-systems. Similarly, the research from the category of L-semilattice is also an important method to investigate the properties of L-semilattice. This paper focuses on the limit the-ory of the category of L-semilattice,which include all L-semilattices and all the actin-preserving maps between them.We give the representation of product and coproduct, equalizer and coequalizer, pushout and pullback in detail, and prove that the category of L-semilattice is an cartesian closed category in the adjoint theory of L-semilattice.Injective modules, projective modules and flat modules are three basic mod-ules, and also the important research content in modules theory. We define the injective L-semilattice based on L-semilattice, and discuss the relevant proper-ties of injective L-semilattice, obtain the necessary and sufficient conditions for the L-semilattice to be injective L-semilattice. Giving a brief introduction of the properties of the free L-semilattice, we prove that a free L-semilattice is a projective L-semilattice. The study of flatness is not only the hot research area in S-systems and module theory, but also the important research direction of L-semilattice. The paper prove that a pullback flat L-semilattice,the coproduct of flat L-semilatticeis and a projective L-semilattice are flat L-semilatticeis respec-tively...