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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>
 +)]