| We study the properties of the algebra of conditional logic, which is a non-commutative regular extension of Boolean logic to three truth values, and the properties of partially complemented distributive lattices, which are distributive lattices with a unary operation and some additional structure. In particular, we show that the variety of partially complemented distributive lattices satisfies the congruence extension and amalgamation properties while the variety generated by the three-element algebra associated to conditional logic does not. We call the latter variety the variety of C-algebras and denote its generator by C. In the absence of the amalgamation property, we study the amalgamation class of the variety of C-algebras and show that, for certain algebras, membership in the amalgamation class is equivalent to satisfying a property known in the literature as property Q. Furthermore, we study properties of the set of all homomorphisms from a C-algebra A to C, which we denote by Hom(A, C). We define a partial order on Hom(A, C) and show that if two elements in the poset Hom(A, C) have an upper bound in Hom(A, C), then they must also have a greatest lower bound in Hom(A, C). We show that given an arbitrary, finite, lower-bounded poset P having this property, we can construct a finite C-algebra whose poset Hom(A, C) is isomorphic to P. This result is then used to establish a dual representation between a category whose objects are precisely these finite, lower-bounded posets and a category whose objects are finite C-algebras. |
