Fractional latin squares, simplex algebras, and generalized quotients

Abstract

Recently the authors defined the concept of a weighted quasigroup, and showed that each weighted quasigroup is the amalgamation of a quasigroup. Similar results were obtained for symmetric and other types of quasigroups. Here we first introduce the closely related concept of a fractional latin square and show that every fractional latin square is the fractional amalgamation of a latin square; we also show that every symmetric fractional latin square is the fractional amalgamation of a symmetric latin square; and we obtain some further similar results about some other types of fractional latin squares. We then introduce the concept of a simplex algebra, and we show that our results about weighted quasigroups and fractional latin squares can be given a very natural geometric formulation in terms of projections of simplex algebras. We also show how our results can be viewed as statements concerning a generalization of the standard quotient construction defined on an algebraic system.