Algebraic syntax
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Algebraic syntax is a theory of syntax developed by Michael Brame as an alternative to Transformational-generative grammar. Brame formulated an algebra (technically a nonassociative groupoid with inverses) of lexical items (words and phrases), or lexes for short. A lex is a string representation of a word or phrase together with a string of directed types. A directed type is a symbol representing a syntactic type together with a direction (up, down, left, right) usually given by an arrow beside or above the symbol. In this article left and down arrows will be placed to the left and right and up arrows to the right of symbols.
Lexical composition of two lexes is performed by concatenating the phonetic or orthographic representations and composing the directed type strings. Thus [A, B] [C, D] = [AC, BD]. In our groupoid of directed type strings we define X→←X = X↑↓X = ←X↓X = X↑X→ = 1 for all X so that these strings "cancel."
With these definitions we can consider the subgroupoid generated by a lexicon of primitive lexes. For example, our lexicon might contain the words [We, ←SV→], [went, ←VN→], and [home, ←N], from which we can construct [We, ←SV→] [went, ←VN→] [home, ←N] = [We went home, ←S]. Given a correct lexicon to begin with, the theory of algebraic syntax claims that the grammatical sentences will be precisely those with directed type ←S.
[Need to add explanation of how up and down arrows are used, as well as recursion to show that a lexicon can generate an infinite grammar and currying to deal with lexes that take multiple complements.]