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boolean_algebras_with_operators [2010/07/29 15:23] (current)
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 +=====Boolean algebras with operators=====
 +
 +Abbreviation: **BAO**
 +
 +====Definition====
 +A \emph{Boolean algebra with operators} is a structure $\mathbf{A}=\langle A,\vee,0,
 +\wedge,1,\neg,f_i\ (i\in I)\rangle$ such that
 +
 +$\langle A,\vee,0,\wedge,1,\neg\rangle$ is a Boolean algebra
 +
 +$f_i$ is \emph{join-preserving} in each argument:  
 +$f_i(\ldots,x\vee y,\ldots)=f_i(\ldots,x,\ldots)\vee f_i(\ldots,y,\ldots)$
 +
 +
 +$f_i$ is \emph{normal} in each argument:  $f_i(\ldots,0,\ldots)=0$
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean algebras with operators of the same signature.
 +A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves all the operators:
 +
 +$h(f_i(x_0,\ldots,x_{n-1}))=f_i(h(x_0),\ldots,h(x_{n-1}))$
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  |decidable |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  |undecidable |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  |unbounded |
 +^[[Congruence distributive]]  |yes |
 +^[[Congruence modular]]  |yes |
 +^[[Congruence n-permutable]]  |yes, $n=2$ |
 +^[[Congruence regular]]  |yes |
 +^[[Congruence uniform]]  |yes |
 +^[[Congruence extension property]]  |yes |
 +^[[Definable principal congruences]]  |no |
 +^[[Equationally def. pr. cong.]]  |no |
 +^[[Amalgamation property]]  |yes |
 +^[[Strong amalgamation property]]  |yes |
 +^[[Epimorphisms are surjective]]  |yes |
 +
 +
 +====Subclasses====
 +[[Modal algebras]]
 +
 +[[Boolean monoids]]
 +
 +
 +====Superclasses====
 +[[Boolean algebras]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]
 +
 +
 +
 +