# Differences

This shows you the differences between two versions of the page.

— |
boolean_algebras_with_operators [2010/07/29 15:23] (current) |
||
---|---|---|---|

Line 1: | Line 1: | ||

+ | =====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> | ||

+ | )] | ||

+ | |||

+ | |||

+ | |||

+ | |||

Trace: