Title: Two-option magmas: Binary Operations on The Natural Numbers Spanned By Their Addition and Multiplication
Speaker: Majed Zailaee (Ohio University)
Abstract: Binary operations on the natural numbers spanned by their addition and multiplication. Let S be any set and $\ast$ and $\circ$ be two arbitrary operations on S. An operation $\star$ on $S$ is said to be a {$\it$ two-option operation spanned by $\ast$ and $\circ$} if for all $a,b \in S$, $a \star b$ $\in \{a \ast b, a\circ b \}$. Any two-option operation may be represented by a graph having the elements of S are vertices and such that there is an edge between a and be precisely when $a \star b = a \ast b$. Two-option operations were motivated by graph magmas and two valued magmas studied earlier in other projects. We are interested in learning what associative operations may be spanned by two given operations $\ast$ and $\circ$. Interestingly, $\ast$ and $\circ$ need not be associative themselves to yield $\star$ associative. As an initial experiment we aim to produce an exhaustive list of associative two option operations on the set of natural numbers for $\ast$ and $\circ$ being, respectively, the usual addition and multiplication of natural numbers.
This is a preliminary report on work in progress with S. López-Permouth and R. Muhammad.
