The following dissertation investigates algebraic frames. Formally speaking, a frame is a complete lattice which satisfies a strengthened distributive law where finite infima distribute over arbitrary suprema. In particular, we are interested in focussing on a certain space associated with an algebraic frame: the space of minimal prime elements. In the first half of the dissertation we will investigate different interesting properties of these topological spaces in terms of the algebraic properties of the frame. In one of our main results we state internal conditions of an algebraic frame which will ensure its minimal prime element space is compact.
In Chapter 5 we will describe the radical of an algebraic frame. This is a generalization in context to the frame of radical ideals of a commutative ring with identity. We will demonstrate that the radical of an algebraic frame is an algebraic frame.
The last part of the dissertation focuses on extensions of algebraic frames. We will generalize the notions of rigid extension, r-extension and r*-extension which are known in the theory of lattice-ordered groups. Our main result will characterize rigid extensions in several ways. We will answer the following question: “Which type of extensions between two algebraic frames will ensure a homeomorphism between their corresponding minimal prime element spaces?” This question had been looked at and answered for lattice-ordered groups by Conrad and Martinez in [4] and later by McGovern in [17]. We will also provide an important example from the theory of rings of continuous functions. In this example, we will construct an extension of algebraic frames which will demonstrate that an r*-extension and an r-extension are two different concepts. In the end we will provide several open questions which may lead to future study.