# Differences

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basic_logic_algebras [2010/07/29 15:14]
jipsen created
basic_logic_algebras [2019/11/17 13:33] (current)
pnotthesamejipsen
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-f+=====Basic logic algebras=====
+
+Abbreviation: **BLA**
+
+====Definition====
+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
+[[bounded lattice]]
+
+$\langle A,\cdot ,1\rangle$ is a [[commutative monoid]]
+
+$\to$ gives the residual of $\cdot$:  $x\cdot y\leq z\Longleftrightarrow y\leq x\to z$
+
+prelinearity:  $( x\to y) \vee ( y\to x) =1$
+
+BL:  $x\cdot(x\to y)=x\wedge y$
+
+Remark:
+The BL identity implies that the lattice is distributive.
+
+====Definition====
+A \emph{basic logic algebra} is a [[FLe-algebra]] $\mathbf{A}=\langle +A,\vee ,0,\wedge ,1,\cdot ,\to \rangle$ such that
+
+linearity:  $( x\to y) \vee ( y\to x) =1$
+
+BL:  $x\cdot (x\to y)=x\wedge y$
+
+Remark:
+The BL identity implies that the identity element $1$ is the top of the lattice.
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be basic logic algebras.
+A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:
+
+$h(x\vee y)=h(x)\vee h(y)$, $h(1)=1$, $h(x\wedge +y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x\cdot +y)=h(x)\cdot h(y)$, $h(x\to y)=h(x)\to h(y)$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+^[[Classtype]]  |variety |
+^[[Equational theory]]  |decidable |
+^[[Quasiequational theory]]  | |
+^[[First-order theory]]  | |
+^[[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]]  | |
+^[[Equationally def. pr. cong.]]  |no |
+^[[Amalgamation property]]  | |
+^[[Strong amalgamation property]]  | |
+^[[Epimorphisms are surjective]]  | |
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &2\\ +f(4)= &5\\ +f(5)= &10\\ +f(6)= &23\\ +f(7)= &49\\ +f(8)= &111\\ +\end{array}$
+
+The number of subdirectly irreducible BL-algebras of size $n$ is $2^{n-2}$.
+
+
+====Subclasses====
+[[MV-algebras]]
+
+[[Heyting algebras]]
+
+
+====Superclasses====
+[[Generalized basic logic algebras]]
+
+[[FLew-algebras]]
+
+
+====References====
+

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