Syntax of infix notation declarations.
InfixNotation is used to treat a composite box structure as an infix operator. InfixNotation requires both a composite box object which will represent the infix operator and a symbol which will be the full form head of the expression. A simple parallel of this duality in Mathematica is that the infix notation + has the full form head Plus.
This declares that the composite object should act as the infix form of Join.
The infix notation both parses input and formats output.
An advantage of using InfixNotation over that of using Notation to define an infix operator is that InfixNotation parses an expression into a flat internal form without evaluation. Although this is a subtle difference, it is an important one (see ring operations and parsing without evaluation.)
InfixNotation parses input to a flat expression.
The function InfixNotation can be compared to using the infix form of functions ~~ (cf. The Mathematica Book 2.1.3)