Amalgamation Property And Lattices Of Ideals In The Relative Ockham Algebras

In this thesis, we consider two classes of relative Ockham algebras, namely theclass of extended Ockham algebras and the class of the balanced pseudocomple-mented Ockham algebras. An extended Ockham algebra (L;∧,∨, f, k, 0, 1) is abounded distributive lattice (L;∧,∨, 0, 1) with a dual endomorphism f and anendomorphism k such that f and k commute. A particular subclass of the classof extended Ockham algebras is the class of e₂M-algebras in which f²=id_Land k²=id_L. We say that an algebra (L;∧,∨,* , f, 0, 1) is a balanced pseu-docomplernented Ockham algebra (shortly, bpO-algebra) if (L;∧,∨,* , 0, 1) is adistributive pseudocomplemented algebra with a dual endomorphism f suchthat f(x^*)= x^** and [f(x)]^*=f²(x) for every x∈L.In chapter 2, we continue Blyth and Fang's work on e₂ M-algebras. We payour considerable attention to the amalgamation property in the class of e₂ M-algebras. The main result is to prove that there are only nine subvarieties ofthe variety of e₂ M-algebras, each of which is generated by single subdirectlyirreducible algebra, have the amalgamation property.In chapter 3, we discuss mostly the properties on lattices of ideals of bpO-algebras (L;∧,∨,* , f, 0, 1). For a bpO-algebra L, we let I(L) and KI(L) bethe lattice of all ideals of L and that of all kernel ideals of L, respectively. Wealso letI₀={x∈L|((?)α∈I)x＞a^*}and I^?={x∈L|((?)∈I)x∧i=0}and letC^*(KI(L))={R_I|I∈KI(L)}in which R_I is a congruence on L that is defined by(x,y)∈R_I(?)((?)i∈(?)I)x∨i=y∨i.The main results we obtained in this chapter are as follows:(1) Let L∈bpO. If I is a kernel ideal of L then(â…°) (x,y)∈θ(I)(?)((?)a, b∈I) (x∨a) b^*=(y∨a)∧b^*;(â…±)θ(I)=θ_lat(I)∨θ_lat(I₀). (2) If L∈bpO then, (KI(L);∨,∧,^*) is a p-sublattice of (I(L);∨,∧) whereI∧J=I∩J and I∨J={x∈L|((?)i∈I)((?)j∈J)x≤i∨j},and the operation^* is defined by I^*={x∈L|((?)i∈I)x∧i=0}.(3) If L∈bpO then, (KI(L);^*) is a Stone algebra if and only if for anyI∈KI(L), I^* is a principal kernel ideal of L.(4) If L∈bpO, then (C^*(KI(L));^*) is a p-sublattice of ConL whereR_I∨R_J=R_I∨J and R_I∧R_J=R_I∧J,and the operation^* is given by (R_I)^*=R_I^*.