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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> | ||
+ | )] |
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