# Differences

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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>
+)]
+
+
+
+

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