α-filters In Residuated Lattices And Their Topological Space

Residuated lattice,which was first introduced by American scholars Ward and Dilworth in 1939,is a very basic and important algebraic structure.Many kinds of non-classical logical algebras,such as MTL-algebra,BL-algebra,Heyt-ing algebra and MV-algebra,can be constructed based on residuated lattices.The filters theory of the logical algebras plays the essential role in studying al-gebraic structure.The aim of this paper is to study α-filters and prime α-filter spaces in residuated lattices.The main contents are as follows:Firstly,we introduce the concepts of co-annihilators and α-filters in residu-ated lattices,and investigate some related properties of them.We also discuss some relations between α-filters and maximal filters,Boolean filters,obstinate filters.In addition,we give an extention of a filter F to become an α-filter E(F),and we get some characterizations of α-filters.We answer the open problem giv-en in[Haveshki M.et al.,On α-filters of BL-algebras,J.of Intell.and Fuzzy Sys.,2015,28:373-382].In fact,we prove that there is no non-trivial α-filter in linear residuated lattice L.So Fa(L)is a trivial structure when L is linear.However,Haveshki studied the lattice structure of Fa(L)in a BL-chain,clearly some results obtained by Haveshki were trivial.But we study the structure of Fa(L)in a general residuated lattice and we can explain that Fa(L)is not a trivial structure,and so some results we get are non-trivial.Finally,we study some topological properties of space of prime α-filters in a residuated lattice.We also give some equivalent conditions for the space to become a T1-space and a Hausdorff space.We get the following results:(1)If a residuated lattice L contains only one co-atom,then L has no any non-trivial α-filter.(2)Let F be a maximal filter of a residuated lattice L such that E(F)≠L.Then F is an α-filter of L.(3)If F and G are filters of a residuated lattice L,then E(F)∩ E(G)=E(F∩G).(4)Let L be a residuated lattice.Then(Fa(L),∩,∨E,*,{1},L)is a pseu-docomplemented lattice,where F ∨E G=E(F V G),F*=F丄,for anyF,G∈Fα(L).(5)Let L be a residuated lattice and F,G J Fα(L).Define F→ G={x ∈ L|<x>∩F(?)G}.Then(Fa(L),∩→E,{1},L)forms a complete Heyting algebra.(6)Let L be a residuated lattice.Then the set Pa(L)of all prime α-filters is a T1-space if and only if every prime α-filter is a maximal prime α-filter if and only if every prime α-filter is a minimal prime α-filter.