Ms-algebras With Double Demi-Pseudocomplementation

In this thesis,we investigate MS-algebras with double demi-pseudocomplement-ation,denoted by ddpMS-algebras；namely those algebras(L；∧,∨,°,*,+,0,1)(briestly(L;°,*,+,))of type〈2,2,1,1,0,0〉in which(L；°)is an MS-algebra,(L；*,+)is a double demi-p algebra,and the unary operations°,*and+are linked by the following identities:(1)x°*=x°+=x∞;(2)x*°=x*+=x**；(3) x+°=x+*=x++.Here we shall characterize the properties of congruences on a ddpMS-algebra and the subdirectly irreducible algebras.We particularly show that if L is a subdirectly irreducible ddpMS-allgebra then the lattice ConL of congruences on L reduces to a two-element chain or three-element chain.Also, we give a complete description of subdirectly irreducible ddpMS-algebras via Priestley topological duality.The main results we obtained im this thesis are given as follows:(A)[Theorem3.2.2]Let(L;°,*,+)be a ddpMS-algebra and a,b∈L with a≤b. Thenθ(a,b)=θlat(a,b)∨θlat(b°,a°)∨θlat(b*,a*)∨θlat(b+,a+).(B)[Theorem3.2.3]If(L；°.*,+)is a ddpMs-algebra,then for any a,b∈L, we have(1)θa∧θa°=ω；(2)θa∧θb=θab;(3)θa=ω(?)a≥a°,where θa is a lattice congruence on L defined by ((?)x,y∈L)(x,y)∈θa(?)x∧a=y∧a and x∨a°=y∨a°.(C)[Theorem3.2.7]Let(L；°,*,+)be a subdirectly irreducible ddpMS-algebra, then the congruence lattice Con L reduces to the chain:ω(?)Φ∧G (?) τ.(D)[Theorem3.4.2]Up to isomorphism,there are precisely36non-trivial sub-directly irreducible ddpMS-algebras.