Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations

A partial algebra construction of Gr\"atzer and Schmidt from "Characterizations of congruence lattices of abstract algebras" (Acta Sci. Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to a well-known fact that every finite distributive lattice is representable, seen as a special case of the Finite Lattice Representation Problem. The construction of this proof brings together Birkhoff's representation theorem for finite distributive lattices, an emphasis on boolean lattices when representing finite lattices, and a perspective based on inequalities of partially ordered sets. It may be possible to generalize the techniques used in this approach. Other than the aforementioned representation theorem only elementary tools are used for the two theorems of this note. In particular there is no reliance on group theoretical concepts or techniques (see P\'eter P\'al P\'alfy and Pavel Pud\'lak), or on well-known methods, used to show certain finite lattice to be representable (see William J. DeMeo), such as the closure method

A short note on a Bernstein-Bezier basis for the pyramid

We introduce a Bernstein-Bezier basis for the pyramid, whose restriction to the face reduces to the Bernstein-Bezier basis on the triangle or quadrilateral. The basis satisfies the standard positivity and partition of unity properties common to Bernstein polynomials, and spans the same space as non-polynomial pyramid bases in the literature.Comment: Submitte