Square Image Of Symbo (⊏) As Used by Barendregt in "The Lambda Calculus - Its Syntax and Semantics"

Question

Barendregt (ISBN 978 1 84890 066 0) discusses "complete partial orderings" in his book "The Lambda Calculus - Its Syntax and Semantics". In this book in 1.2.1 he states "Let D = (D, ⊏) be a poset [partially ordered set] with a reflexive ⊏". But what IS a ⊏ in the first place? I've figure that it's pronounced "square image of".

But he does not say what he means by the "square image of".

This symbol can be used by him in statements such as "For all x, y in X, there is a z in X [x ⊏ z and y ⊏ z]" But again I can't figure out what the ⊏ actually means. That definition is a mostly English translation of a description of a directed set of some sort.

This symbol can be negated (appear with a line through it). It might be similar to a supremum (which looks like ⊏ but rotated 90 deg. to the left/counter clockwise).

Does anyone know vaguely what it might mean? It's used throughout but not defined nor indexed in the book.

Answer 1

The symbol “⊏” has no generally defined meaning (the way that symbols like “<” and “⊂” have predefined meanings, even though you can give them your own definitions). It has whatever meaning someone chooses to give it in her or his definition in some context.

It has apparently been used as a relation symbol, typically for a partial ordering, possibly as a generic symbol for such a relation, or for some specific relation. Its shape can be interpreted as an angular variant of the subset symbol “⊂” It has been used often enough in printed works, so that it was encoded in Unicode as U+228F SQUARE IMAGE OF.

The Unicode “name” (alphanumeric identifier) SQUARE IMAGE OF does not correspond to typical use of the symbol, and it is unclear where it comes from. The word SQUARE reflects the angular form of the symbol, but the symbol does not look the least similar to “⊷” U+22B7 IMAGE OF (which suggests the relation of being the image of an element in a mapping).

(I found this question and the useful comments, which I used when writing this answer, while looking for information for the definition of natural-language names for this symbol and its relatives. It seems that such naming should not be based on the Unicode “name” but rather the idea “angular relational operator”.)

Answer 2

By definition, a "poset" is a set that is equipped with a reflexive transitive relation. If the poset is $(D, ⊏)$, then $D$ is the name of the set, and $⊏$ is the name of the relation.

There is no further answer to "what is $D$"; it is just some arbitrary set we will think about. And there is no further answer to "what is $⊏$"; it is just some arbitrary reflexive transitive relation on the set $D$.

Barendregt is using the symbol $⊏$ here to emphasize that the relation has no other properties that are relevant to the discussion. If he had used $≤$, for example, it might appear that he wanted you to understand $D$ as a set of numbers and $≤$ as the less-than-or-equal relation. But Barendregt wants you to consider the more general situation that includes that one as a special case.

One often sees $\preceq$ used similarly.