arXiv subjects
Publications and source records attributed to Joseph Oliveira.
In $\mathcal L$, the semilattice of faces of an $n$-cube, we count the number of automorphisms of $\mathcal L$ that fix a given subalgebra -- either pointwise or as a subalgebra. By using Möbius inversion we get a formula for the number of derangements on the $n$-cube in terms of the Möbius function on the lattice of MR-subalgebras. We compute this Möbius function.
We discuss the group of automorphisms of a general MR-algebra. We develop several functors between implication algebras and cubic algebras. These allow us to generalize the notion of inner automorphism. We then show that this group is always isomorphic to the group of inner automorphisms of a filter algebra.
We show that the variety of symmetric implication algebras is generated from cubic implication algebras and Boolean algebras. We do this by developing the notion of a locally symmetric implication algebra that has properties similar to cubic implication algebras and provide a representation of these algebras as subalgebras of a product of a cubic implication algebra and an implication algebra. We then show that every symmetric implication algebra is covered by a locally symmetric implication algebra.
We consider relationships between cubic algebras and implication algebras. We first exhibit a functorial construction of a cubic algebra from an implication algebra. Then we consider an collapse of a cubic algebra to an implication algebra and the connection between these two operations. Finally we use the ideas of the collapse to obtain a Stone-type representation theorem for a large class of cubic algebras.
We construct free cubic implication algebras with finitely many generators, and determine the size of these algebras.
Let $B$ be a finite Boolean algebra. Let $\mathcal A$ be the partial order of all implication sublattices of $B$. We will compute the Möbius function on $\mathcal A$ in two different ways.
We consider two constructions of an envelope for a finite locally distributive strong upper semilattice. The first is based on Birkhoff's representation of finite distributive lattices and the second on valuations on lattices. We show that these produce isomorphic envelopes.
We show that the class of Metropolis-Rota implication algebras can be given a universal axiomatization using an operation closely related to composition in oriented matroids. Lastly we describe the role of our new operation in the collapse of an MR-algebra.
We consider several distinct characterizations of finite implication algebras. One of these leads to a new characterization of Boolean polymatroids.
In this paper we discuss the inclusion ordering on the filters of a filter algebra, a special type of Metropolis-Rota algeba. Using embeddings into interval algebras we show that the notion of "untwisted" gives rise to a congruence relation on the group of g-filters. We also show that there is a natural reflection operator on the class of filters with an easily definable enveloping cubic subalgebra.