Optimization Over Pseudo-Boolean Lattices M.Hosseinyazdi, A.Hassankhani and M.Mashinchi Faculty of Math. and Computer Sciences Kerman University, Kerman, IRAN Abstract In this paper, first we will find the so- lution of the system A ∗ X ≤ b, where A, b are the known suitable matrices and X is the unknown matrix over a pseudo- Boolean lattice. Then its application to find the solution of some fuzzy linear sys- tems as well as finding the solution of the optimization problem Z = max{C ∗X |A∗ X ≤ b} is discussed. Key-words : pseudo-Boolean lattice, lin- ear programming, fuzzy linear systems, optimization. 1 Introduction and prelim- inaries Linear and combinatorial optimization have been studied by many authors [5]. Op- timization over residuted, lattice-ordered commutative monoid is studied in [5]. On the other hand in many applications, one need to find the solution of fuzzy linear systems of equations and inequalities over a bounded chain in [3]. In this paper we replace a bounded chain by any pseudo- Boolean lattice R, which is recently stud- ied in [1]. Then by using the approach in [5] we will solve linear system A ∗ X ≤ b, over R. The method given here is very easy to be applied for solving the fuzzy linear systems studied in [3]. Definition 1.1. [1,5] A bounded lattice (L, ≤) is called pseudo-Boolean if for all a, b ∈ L, there exists c ∈ L such that a ∧ x ≤ b ⇔ x ≤ c ∀x ∈ L. If such element c exists, then it is unique and will be denoted by b : a. For the following remark see [5]. Remark 1.2. (i) Every finite distribu- tive lattice is pseudo-Boolean. (ii) In a Boolean lattice B, one can see that b : a = b ∨ a ∗ . Hence B is pseudo- Boolean. (iii) In general, a pseudo-Boolean lattice may not be Boolean. For example con- sider a bounded chain (L, ≤). Then a ∧ b = min(a, b), a ∨ b = max(a, b) and