Abstract
A $\mu$-way $k$-homogeneous Latin trade was defined by Bagheri Gh, Donovan, Mahmoodian (2012), where the existence of $3$-way $k$-homogeneous Latin trades was specifically investigated. We investigate the existence of a certain class of $\mu$-way $k$-homogeneous Latin trades with an idempotent like property. We present a number of constructions for $\mu$-way $k$-homogeneous Latin trades with this property, and show that these can be used to fill in the spectrum of $3$-way $k$-homogeneous Latin trades for all but $196$ possible exceptions.
Highlights
A partial Latin square of order m, T = [t(r, c)], is an m × m array of cells with each cell either filled with an element t(r, c) of Ω or left empty, such that each symbol of Ω appears at most once in each row, and at most once in each column
We investigate the existence of a certain class of μ-way k-homogeneous Latin trades with an idempotent like property
We present a number of constructions for μ-way k-homogeneous Latin trades with this property, and show that these can be used to fill in the spectrum of 3-way k-homogeneous Latin trades for all but 196 possible exceptions
Summary
A partial Latin square of order m, T = [t(r, c)], is an m × m array of cells with each cell either filled with an element t(r, c) of Ω (a set of m symbols) or left empty, such that each symbol of Ω appears at most once in each row, and at most once in each column. The spectrum of μ-way homogeneous Latin trades of order m, Smμ , is the set of values of k such that there exists a (μ, k, m)-Latin trade. The spectrum of idempotent μ-way homogeneous Latin trades of order m, ISμm, is the set of values of k such that there exists an idempotent (μ, k, m)-Latin trade. We show there exists 3-way k-homogeneous Latin trades of order m with 4 k m for all but a finite list of possible exceptions
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