The 0-distributivity in the class of subalgebra lattices of Heyting algebras and closure algebras

Using Priestley duality, we characterize those Heyting and closure algebras whose subalgebra lattice is 0-distributive (i.e.satisfies xAy = 0 and x A z = 0 => xA(yvz)-O), Key words: Heyting and closure algebras, subalgebra lattice, O-distributivity, congruences on quasi-ordered topological spaces.* Classification: 06D05 Introduction.In [2],[3] and [5], we study the subalgebra lattice of Heyting algebras and closure algebras and characterize those Heyting algebras and closure algebras whose subalgebra lattice is distributive.Besides, our results characterize in the class D of distributive lattices those which are subalgebra lattices of Heyting algebras or closure algebras.In this paper, we extend the class D to the wider class of O-distributive (i.e.lattices which satisfy the following weakening of the distributivity law: x A y = 0 and XAZ = 0 imply xA(yvz)=0).To obtain these results we use a duality between closure algebras and closure spaces and the notion of congruence on quasi-ordered topological spaces.We recall these notions in the first paragraph.§ 1 Recalls 1.1.Definitions, (a) A closure algebra B=(B; A,V, ,~,0,1) is a Boolean algebra (B;A ,V, C ,0,D with a unary operator (closure operator) satisfying (i) 0"=0; (ii) V xeB,x<=x~ = x~ ~; (iii) Vx, yeB,(xvy)" = x"vy".