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+ | =====Distributive lattices with operators===== | ||

+ | |||

+ | Abbreviation: **DLO** | ||

+ | |||

+ | ====Definition==== | ||

+ | A \emph{distributive lattice with operators} is a structure $\mathbf{A}=\langle A,\vee,\wedge,f_i\ (i\in I)\rangle$ such that | ||

+ | |||

+ | $\langle A,\vee,\wedge\rangle$ is a [[distributive lattice]] | ||

+ | |||

+ | $f_i$ is \emph{join-preserving} in each argument: | ||

+ | $f_i(\ldots,x\vee y,\ldots)=f_i(\ldots,x,\ldots)\vee f_i(\ldots,y,\ldots)$ | ||

+ | |||

+ | ==Morphisms== | ||

+ | Let $\mathbf{A}$ and $\mathbf{B}$ be distributive lattices with operators of the same signature. | ||

+ | A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a distributive lattice 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]] | | | ||

+ | ^[[Congruence regular]] | | | ||

+ | ^[[Congruence uniform]] | | | ||

+ | ^[[Congruence extension property]] | | | ||

+ | ^[[Definable principal congruences]] |no | | ||

+ | ^[[Equationally def. pr. cong.]] |no | | ||

+ | ^[[Amalgamation property]] |yes | | ||

+ | ^[[Strong amalgamation property]] |yes | | ||

+ | ^[[Epimorphisms are surjective]] |yes | | ||

+ | |||

+ | |||

+ | ====Subclasses==== | ||

+ | [[Bounded distributive lattices with operators]] | ||

+ | |||

+ | [[Distributive lattice-ordered semigroups]] | ||

+ | |||

+ | |||

+ | ====Superclasses==== | ||

+ | [[Distributive lattices]] | ||

+ | |||

+ | |||

+ | ====References==== | ||

+ | |||

+ | [(Ln19xx> | ||

+ | )] | ||

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