UFL Talk

Outline: 

Decidability

Searching for (counter-) examples

Putting results on the web

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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`
<center>`x*y<=z iff x<=z/y iff y<=x\z`</center>

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
<center>
`s_1=t_1 and ... and s_n=t_n implies s_0=t_0` 
</center>
where the `s_i,t_i`
are terms.

When `n=0`, this reduces to an identity `s_0=t_0`.

An inequality `s<=t` is of course equivalent to the identity
`s^^t=s`.

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, 
<center>
`x.(y vv z) = x.y vv x.z`     `(x vv y).z = x.z vv y.z`
`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`.
</center>

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
`x^-1=x\e`.

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.

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