On the Lattice of Filters of Intuitionistic Linear Algebras

In this paper‎, ‎we investigate the filter theory of Intuitionistic Linear Algebra (IL-algebra‎, ‎in short) with emphasis on the lattice of filters of IL-algebras and relationship between filters and congruences on IL-algebras‎. ‎We characterize the filter generated by a subset and give some related properties‎. ‎The prime filter for IL-algebras is characterized and the prime filter theorem for IL-algebra is established‎. ‎We get that the lattice (F(\textbf{L}),\subseteq ) of filters of an IL-algebra \textbf{L} is algebraic‎, ‎Brouwerian‎, ‎pseudocomplemented and endowed with the structure of Heyting algebra‎. ‎We prove that the lattice of congruences and that of filters of any IL-algebra are isomorphic‎.

Keywords:

IL-algebra‎ , ‎Filter‎ , ‎Prime filter‎ , ‎Congruence‎ , ‎Residuated lattice.‎