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brouwerian_algebras [2010/07/29 15:23] (current)
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+=====Brouwerian algebras=====
+Abbreviation: **BrA**
+
+====Definition====
+A \emph{Brouwerian algebra} is a structure $\mathbf{A}=\langle A, \vee, \wedge, 1, \rightarrow\rangle$ such that
+
+
+$\langle A, \vee, \wedge, 1\rangle$ is a distributive lattice
+with top
+
+
+$\rightarrow$ gives the residual of $\wedge$:  $x\wedge y\leq z\Longleftrightarrow y\leq x\rightarrow z$
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be Brouwerian algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
+homomorphism:
+
+$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$
+
+====Definition====
+A \emph{Brouwerian algebra} is a BL-algebra $\mathbf{A}=\langle A, \vee, \wedge, 1, \cdot, \rightarrow\rangle$ such that
+
+$x\wedge y=x\cdot y$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+^[[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, $n=2$ |
+^[[Congruence e-regular]]  |yes, $e=1$ |
+^[[Congruence uniform]]  |no |
+^[[Congruence extension property]]  |yes |
+^[[Definable principal congruences]]  |yes |
+^[[Equationally def. pr. cong.]]  |yes |
+^[[Amalgamation property]]  |yes |
+^[[Strong amalgamation property]]  |yes |
+^[[Epimorphisms are surjective]]  |yes |
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &1\\ +f(4)= &2\\ +f(5)= &3\\ +f(6)= &5\\ +f(7)= &8\\ +f(8)= &15\\ +f(9)= &26\\ +f(10)= &47\\ +f(11)= &82\\ +f(12)= &151\\ +f(13)= &269\\ +f(14)= &494\\ +f(15)= &891\\ +f(16)= &1639\\ +f(17)= &2978\\ +f(18)= &5483\\ +f(19)= &10006\\ +f(20)= &18428\\ +Values known up to size 49 [Erne, Heitzig, Reinhold (2002)] +\end{array}$
+
+
+====Subclasses====
+[[Generalized Boolean algebras]]
+
+[[Heyting algebras]]
+
+
+====Superclasses====
+[[Distributive lattices]]
+
+[[Basic logic algebras]]
+
+
+====References====
+
+[(Ln19xx>
+)]