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bounded_residuated_lattices [2010/07/29 15:23] (current)
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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]]
 +)]
 +
 +