bounded_residuated_lattices

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=====Bounded residuated lattices=====

Abbreviation: **RLat$_b$**

====Definition====
A \emph{bounded residuated lattice} is a [[residuated lattice]]
that is bounded:

$\bot$ is the least element:  $\bot\vee x=x$

$\top$ is the greatest element:  $\top\vee x=\top$

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be bounded residuated lattices. 
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a residuated lattice homomorphism $h:A\rightarrow B$ that preserves the bounds: 
$h(\bot)=\bot$ and $h(\top)=\top$.

====Examples====
Example 1: 

====Basic results====


====Properties====
^[[Classtype]]                        |variety  |
^[[Equational theory]]                |decidable  |
^[[Quasiequational theory]]           |undecidable  |
^[[First-order theory]]               |undecidable  |
^[[Locally finite]]                   |no  |
^[[Residual size]]                    |unbounded  |
^[[Congruence distributive]]          |yes  |
^[[Congruence modular]]               |yes  |
^[[Congruence $n$-permutable]]        |yes, $n=2$  |
^[[Congruence regular]]               |yes  |
^[[Congruence uniform]]               |no  |
^[[Congruence extension property]]    |yes  |
^[[Definable principal congruences]]  |no  |
^[[Equationally def. pr. cong.]]      |no  |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====Finite members====

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


====Subclasses====
  [[...]] subvariety

  [[...]] expansion


====Superclasses====
  [[...]] supervariety

  [[...]] subreduct


====References====

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

Trace: • first-order_theory • loops • commutative_residuated_lattices • finite_semilattices_with_identity • rings • quasigroups • inverse_semigroups • euclidean_domains • equivalence_relations • semirings