Utral

Utral is an esolang created by User:Yayimhere, to answer her very important question, "What if there was a negative null", and this answer's this question! It is based in Lambda Calculus, however that is only in underlying calculation, and not everything is a function, nor Is the main concept of Utral made specifically with Lambda Calculus in mind.

Etymology

Utral's name comes from the idea of an un-neutral, as null is a sort of neutral value, it literally is nothing after all!

Info

the first, and most basic thing is the only real value/input in Utral, () which when passed to any function, will delete any mentions of the input name it was passed in by within the function, so passing () as an input to this function:

λx.λy. x (y x)
would result in:
λy. y
which is the identity function. 

next is -z where z can be any Utral valid program. -z applies the church integer negation function(However Utral doesnt have church integrals) to whatever z is. the function itself is:

λz. (λa.λb.λf. f a b) ((λp. p (λa.λb. a)) z) ((λp. p (λa.λb. b)) z)

now the last command is <x><y>, where x and y also can be any valid Utral program(they must be evaluated before the command itself is). <x><y> is the same as x y in lambda calculus(as in apply x to y). <x> is interpreted as normal brackets when there is no corresponding <y>.

unlike some may assume, Utral will evaluate and simplify a specific function, just until it no longer can be evaluated nor simplified, and then that will be the final result.

some may also assume that Utral can use normal lambda calculus expression's outside of this syntax, however, that is also not true, as the above is the only valid syntax.