Searching for (counter-) examples
Putting results on the web
A residuated lattice
is an algebra `bfL=<<(L,vv,^^,*,e,\,/)>>` such that `<<(L,vv,^^)>>` is a
lattice, `<<(L,*,e)>>` is a monoid, and for all `x,y,z in L`
We usually write `x*y` as `x.y` and use the convention that,
in the absence of parentheses,
`*` is performed first, followed by `\,/` and then `vv, ^^`.
A quasi-identity is an implication of the form
where the `s_i,t_i`
When `n=0`, this reduces to an identity `s_0=t_0`.
An inequality `s<=t` is of course equivalent to the identity
A class K of universal algebras is a (quasi-) variety if
it is the class
of all algebras that satisfy a set of (quasi-) identities.
The class RL of all residuated lattices is a variety.
An equational basis is given by the monoid and lattice identities together
with, for instance,
`x\(y ^^ z) = x\y ^^ x\z` `(x ^^ y)/z = x/z ^^ y/z`
`x <= x.y/y` `(x/y).y <= x`
`y <= x\x.y` `x.(x\y) <= y`.
Dilworth assumes that `e` is the top element. Such residuated
lattices are called integral.
The class of all integral residuated lattices
is defined by the identity `x^^e=x` and is denoted by IRL.
In the first 3 papers, he also assumes that
`*` is a commutative operation,
which is equivalent to `x\y=y/x`.
In this case it is common to
denote the residuals by `x->y=x\y=y/x`.
CRL is the variety of commutative residuated lattices.
A lattice ordered group is (term equivalent to) a residuated
lattice that satisfies `x.(x\e)=e`.
In this case it is common to write
It follows quite easily that `x^-1.x=e`, `x\e=e/x`,
`x/y=x.y^-1` and `x\y=x^-1.y`.
LG is the variety of l-groups