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In rings: Let $R$ be a ring where idempotents lift modulo the Jacobson radical $J(R)$. Any countable set of orthogonal idempotents in $R/J(R)$ lifts to an orthogonal set of idempotents; but this fails for uncountable sets.

In combinatorics of words: Vaughan Pratt's "crossword problem" is another example. Suppose you have a language consisting of words of length $\kappa$ in the letters $\{0,1\}$, satisfying the following three conditions: (1) The all $0$'s word, and the all $1$'s word, belong to your language. (2) For any two distinct positions in $\kappa$, there is a word with $1$ in the first position and $0$ in the second. (3) If you fill in a $\kappa\times\kappa$ crossword, so that every row and every column is a word in your language, then the main diagonal is also a word.

Pratt asked: Must your language contain all possible words? The answer is yes if $\kappa\leq \aleph_0$, and no otherwise.

In rings: Let $R$ be a ring where idempotents lift modulo the Jacobson radical $J(R)$. Any countable set of orthogonal idempotents in $R/J(R)$ lifts to an orthogonal set of idempotents; but this fails for uncountable sets.

In combinatorics of words: Vaughan Pratt's "crossword problem" is another example. Suppose you have a language consisting of words of length $\kappa$ in the letters $\{0,1\}$, satisfying the following three conditions: (1) The all $0$'s word, and the all $1$'s word, belong to your language. (2) For any two distinct positions in $\kappa$, there is a word with $1$ in the first position and $0$ in the second. (3) If you fill in a $\kappa\times\kappa$ crossword, so that every row and every column is a word in your language, then the main diagonal is also a word.

Pratt asked: Must your language contain all possible words? The answer is yes if $\kappa\leq \aleph_0$, and no otherwise.

In the converse direction:

In algebras: Amitsur's theorem says that if $R$ is a nil algebra over an uncountable field, then the polynomial ring $R[X]$ is nil as well. Agata showed that this fails for nil algebras over countable fields.

In rings: Let $R$ be a ring where idempotents lift modulo the Jacobson radical $J(R)$. Any countable set of orthogonal idempotents in $R/J(R)$ lifts to an orthogonal set of idempotents; but this fails for uncountable sets.

In comonoids: Vaughan Pratt's "crossword problem" is another example. Suppose you have a language consisting of words of length $\kappa$ in the letters $\{0,1\}$, satisfying the following three conditions: (1) The all $0$'s word, and the all $1$'s word, belong to your language. (2) For any two distinct positions in $\kappa$, there is a word with $1$ in the first position and $0$ in the second. (3) If you fill in a $\kappa\times\kappa$ crossword, so that every row and every column is a word in your language, then the main diagonal is also a word.

Pratt asked: Must your language contain all possible words? The answer is yes if $\kappa\leq \aleph_0$, and no otherwise.

In rings: Let $R$ be a ring where idempotents lift modulo the Jacobson radical $J(R)$. Any countable set of orthogonal idempotents in $R/J(R)$ lifts to an orthogonal set of idempotents; but this fails for uncountable sets.

In combinatorics of words: Vaughan Pratt's "crossword problem" is another example. Suppose you have a language consisting of words of length $\kappa$ in the letters $\{0,1\}$, satisfying the following three conditions: (1) The all $0$'s word, and the all $1$'s word, belong to your language. (2) For any two distinct positions in $\kappa$, there is a word with $1$ in the first position and $0$ in the second. (3) If you fill in a $\kappa\times\kappa$ crossword, so that every row and every column is a word in your language, then the main diagonal is also a word.

Pratt asked: Must your language contain all possible words? The answer is yes if $\kappa\leq \aleph_0$, and no otherwise.

Vaughan Pratt's "crossword problem" is another example. Suppose you have a language consisting of words of length $\kappa$ in the letters $\{0,1\}$, satisfying the following three conditions: (1) The all $0$'s word, and the all $1$'s word, belong to your language. (2) For any two distinct positions in $\kappa$, there is a word with $1$ in the first position and $0$ in the second. (3) If you fill in a $\kappa\times\kappa$ crossword, so that every row and every column is a word in your language, then the main diagonal is also a word.

Pratt asked: Must your language contain all possible words? The answer is yes if $\kappa\leq \aleph_0$, and no otherwise.

In rings: Let $R$ be a ring where idempotents lift modulo the Jacobson radical $J(R)$. Any countable set of orthogonal idempotents in $R/J(R)$ lifts to an orthogonal set of idempotents; but this fails for uncountable sets.

In comonoids: Vaughan Pratt's "crossword problem" is another example. Suppose you have a language consisting of words of length $\kappa$ in the letters $\{0,1\}$, satisfying the following three conditions: (1) The all $0$'s word, and the all $1$'s word, belong to your language. (2) For any two distinct positions in $\kappa$, there is a word with $1$ in the first position and $0$ in the second. (3) If you fill in a $\kappa\times\kappa$ crossword, so that every row and every column is a word in your language, then the main diagonal is also a word.

Pratt asked: Must your language contain all possible words? The answer is yes if $\kappa\leq \aleph_0$, and no otherwise.