Pseudocomplemented Symmetric Extended Ockham Algebras

A pseudocomplemented extended Ockham algebra is an algebra of type<2,2,1,1,1,0,0>where(L;∧,∨,f,k,0,1)is an extended Ockham algebra,(L;∧,∨,f,k,0,1)is a pseudocomplemented algebra and the operations x(?)f(x),x(?)k(x)and x(?)x*commute each other.Specially,if(L;∧,∨,f,k,0,1)is an e2,0K1,1-algebra(i.e.an extended Ockham algebra in which the dual endomorphism f and the endomorphism k are satisfied the identities f3 =f and k2 = idL)then(L;∧,∨,f,k,*,0,1)is called pseudocomplemented e2,0K1,1-algebra(shortly,pe2,0K1,1-algebra).In this thesis,we investigate properties of congruences on such an algebra and the subdirectly irreduciblity.The main results we obtained are as follows:[Theorem 3.2.2]Let(L;°,+,*)∈e2,0K1,1.Then the following statements hold:(1)G=ω(?)(L;*)is boolean;(2)Φ=ω(?)((?)∈L)x**=x°°=x+-;(3)Φ=G(?)((?)∈L)x**=x°°.where Φ and G are the congruences on L defined as the following:(x,y)∈Φ=ω(?)x°=y°;(x,y)∈G=ω(?)x*=y*;[Theorem 3.2.3]If(L;°,+,*)∈pe2,0K1,1 and a,b∈L with a≤b,thenθ(a,b)= θ。(a,b)∨θ。(a+,b+)∨θ*(a,b)∨θ(a+,b+)= θlat(a,b)∨θlat(b°,a°)∨θlat(a°°,b°°)∨ θlat(a+,b+)∨θlat(b+°,a+°)∨θlat(a+°°,b+°°)∨θlat(at(a ∧b)*,1)∨θlat((a*∧b)*+,1).[Theorem 4.1.4]Let L g pe2,0K1,1.Then L is properly subdirectly irreducible if and only if Con L ≈{ω}(?)[G,Φ](?){τ},where ω and τ denote the equality relation and the universal relation,respectively.[Theorem 4.2.2]If L∈ pe2,0K1,1 be subdirectly irreducible,then | S(L)|≤16.Where S(L)= {x∈L|x°°=x}.