# Differences

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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]]
+)]
+
+

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