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basic_logic_algebras [2010/07/29 15:14]
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basic_logic_algebras [2019/11/17 13:33] (current)
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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==== 
 +