I have a question about a combinatorial design very similar to a Latin Square, which is arising out of an open problem in graph theory. The design is an $n \times n$ matrix whose entries we want to assign colors from $\lbrace 1, 2, \dots, n \rbrace$. A (symmetric) Latin Square is subject to the following constraints:

1. The matrix should be symmetric.

2. No row or column contains the same color more than once.

In my problem, we will want a symmetric $n \times n$ matrix, but we will allow some of the matrix entries to be identified into symmetric blocks. In other words, suppose there is an equivalence relation $\mathcal{A}$ on the entries $(a_{i,j})$ such that if $A \in \mathcal{A}$ and $a_{i,j} \in A$, then $a_{j,i} \in A$. The problem I am curious about is, if given an $n$ and an arbitrary such $\mathcal{A}$, is there an $n \times n$ matrix satisfying the following constraints:

1. The matrix is symmetric.

2. If $a, b \in A$ for some $A \in \mathcal{A}$, then the entries are assigned the same color, and

3. If two entries $a, b$ are in the same row or column as one another and they are the same color, then there is an $A \in \mathcal{A}$ such that $a, b \in A$.

In other words, it's a symmetric Latin Square 'up to equivalence by $\mathcal{A}$.'

Have these objects been studied before (with or without the symmetry condition), or are there any equivalent formulations? I expect that it is possible to construct such a design. I can only prove it in very special cases, though (e.g. where each row or column only 'intersects' at most one non-trivial equivalence class).

I have a question about a combinatorial design very similar to a Latin Square, which is arising out of an open problem in graph theory. The design is an $n \times n$ matrix whose entries we want to assign colors from $\lbrace 1, 2, \dots, n \rbrace$. A (symmetric) Latin Square is subject to the following constraints:

1. The matrix should be symmetric.

2. No row or column contains the same color more than once.

In my problem, we will want a symmetric $n \times n$ matrix, but we will allow some of the matrix entries to be identified into symmetric blocks. In other words, suppose there is an equivalence relation $\mathcal{A}$ on the coordinates $(a_{i,j})$ which satisfies the property that if $A \in \mathcal{A}$ and $a_{i,j} \in A$, then $a_{j,i} \in A$. The problem I am curious about is, if given an $n$ and an arbitrary such $\mathcal{A}$, is there an $n \times n$ matrix satisfying the following constraints:

1. The matrix is symmetric.

2. If $a, b \in A$ for some $A \in \mathcal{A}$, then the entries are assigned the same color, and

3. If two coordinates $a, b$ are in the same row or column as one another and they are the same color, then there is an $A \in \mathcal{A}$ such that $a, b \in A$.

In other words, it's a symmetric Latin Square 'up to equivalence by $\mathcal{A}$.'

Have these objects been studied before (with or without the symmetry condition), or are there any equivalent formulations? I expect that it is possible to construct such a design. I can only prove it in very special cases, though (e.g. where each row or column only 'intersects' at most one non-trivial equivalence class).