=====Integral residuated lattices=====

Abbreviation: **IRL**

====Definition====
An \emph{integral residuated lattice} is a [[residuated lattice]] $\mathbf{L}=\langle L, \vee, \wedge, \cdot, 1, \backslash, /\rangle$ that is

\emph{integral}:  $x\le 1$

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be integal residuated lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash
y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$

====Examples====
Example 1: The negative cone of any l-group, e.g., $\mathbb Z^-$

====Basic results====


====Properties====

^[[Classtype]]                        |variety  |
^[[Equational theory]]                |decidable |
^[[Quasiequational theory]]           |decidable |
^[[First-order theory]]               |undecidable |
^[[Locally finite]]                   |no  |
^[[Residual size]]                    |unbounded  |
^[[Congruence distributive]]          |yes  |
^[[Congruence modular]]               |yes  |
^[[Congruence $n$-permutable]]        |yes  |
^[[Congruence regular]]               |no  |
^[[Congruence $e$-regular]]           |yes  |
^[[Congruence uniform]]               |no  |
^[[Congruence extension property]]    |no  |
^[[Definable principal congruences]]  |no  |
^[[Equationally def. pr. cong.]]      |no |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====Finite members====

$\begin{array}{lr}
  f(1)= &1\\
  f(2)= &1\\
  f(3)= &2\\
  f(4)= &9\\
  f(5)= &49\\
\end{array}$     
$\begin{array}{lr}
  f(6)= &364\\
  f(7)= &\\
  f(8)= &\\
  f(9)= &\\
  f(10)= &\\
\end{array}$


====Subclasses====
[[commutative integral residuated lattices]]

[[bounded integral residuated lattices]]


====Superclasses====
[[residuated lattices]]

[[integral lattice-ordered monoids]]


====References====

[(Ln19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]