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basic_logic_algebras [2010/07/29 18:30]
127.0.0.1 external edit
basic_logic_algebras [2010/09/04 16:47] (current)
jipsen
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====Definition==== ====Definition====
-A \emph{basic logic algebra} or \emph{BL-algebra} is a structure $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,\cdot ,arrow \rangle $ such that+A \emph{basic logic algebra} or \emph{BL-algebra} is a structure $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,\cdot ,\to \rangle $ such that
$\langle A,\vee ,0,\wedge ,1\rangle $ is a $\langle A,\vee ,0,\wedge ,1\rangle $ is a
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$\langle A,\cdot ,1\rangle $ is a [[commutative monoid]] $\langle A,\cdot ,1\rangle $ is a [[commutative monoid]]
-$arrow $ gives the residual of $\cdot $:  $x\cdot y\leq z\Longleftrightarrow y\leq xarrow z$+$\to$ gives the residual of $\cdot $:  $x\cdot y\leq z\Longleftrightarrow y\leq x\to z$
-prelinearity:  $( xarrow y) \vee ( yarrow x) =1$+prelinearity:  $( x\to y) \vee ( y\to x) =1$
-BL:  $x\cdot(xarrow y)=x\wedge y$+BL:  $x\cdot(x\to y)=x\wedge y$
Remark: Remark:
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====Definition==== ====Definition====
A \emph{basic logic algebra} is a [[FLe-algebra]] $\mathbf{A}=\langle A \emph{basic logic algebra} is a [[FLe-algebra]] $\mathbf{A}=\langle
-A,\vee ,0,\wedge ,1,\cdot ,arrow \rangle $ such that+A,\vee ,0,\wedge ,1,\cdot ,\to \rangle $ such that
-linearity:  $( xarrow y) \vee ( yarrow x) =1$+linearity:  $( x\to y) \vee ( y\to x) =1$
-BL:  $x\cdot (xarrow y)=x\wedge y$+BL:  $x\cdot (x\to y)=x\wedge y$
Remark: Remark:
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==Morphisms== ==Morphisms==
-Let $\mathbf{A}$ and $\mathbf{B}$ be basic logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:Aarrow B$ that is a +Let $\mathbf{A}$ and $\mathbf{B}$ be basic logic algebras.  
-homomorphism: +A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(1)=1$, $h(x\wedge $h(x\vee y)=h(x)\vee h(y)$, $h(1)=1$, $h(x\wedge
y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x\cdot y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x\cdot
-y)=h(x)\cdot h(y)$, $h(xarrow y)=h(x)arrow h(y)$+y)=h(x)\cdot h(y)$, $h(x\to y)=h(x)\to h(y)$
====Examples==== ====Examples====
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The number of subdirectly irreducible BL-algebras of size $n$ is $2^{n-2}$. The number of subdirectly irreducible BL-algebras of size $n$ is $2^{n-2}$.
-\hyperbaseurl{http://math.chapman.edu/structures/files/}+
====Subclasses==== ====Subclasses====
[[MV-algebras]] [[MV-algebras]]
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====References==== ====References====
-[(Ln19xx> +
-)]+