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Mathematics Colloquium: Minc’s Algebra and the Binary Partition Polytopes

Mar 28, 2025, 3:30 - 4:30 PM

 

Speaker:  Jim Lawrence, GMU

Title: Minc’s Algebra and the Binary Partition Polytopes

Abstract: Minc and his advisor Etherington studied the identity (a⊕b)⊕(c⊕d) = (a⊕ c) ⊕(b ⊕ d) and related algebraic systems. In this talk a connection between such a system and the binary partition polytopes will be examined. The “connecting tissue” is provided by the smallest lattice that is isomorphic to its own lattice of intervals.

Starting in dimensions 1 and 2 with the interval and the square, the binary partition polytope Bn of dimension n is a polytope which “looks like” the lattice of faces of the binary partition polytope of dimension n−1. A binary partition of a natural number n is a representation of n as a sum of powers of 2. The faces of Bn−1 (and the vertices of Bn) correspond to the binary partitions of 2n. The binary partitions of 4 are 1+1+1+1, 2+1+1, 2+2, 4, corresponding to the four vertices of a square, or to the four faces (the empty set, the segment itself, and its two vertices) of a line segment.

Some open problems will be described.

 

Time:  Friday, March 28, 3:30pm – 4:20pm

Place:  Exploratory Hall, room 4106

Zoom and In-person

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