function_rings - MathStructures

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=====Function rings=====

Abbreviation: **FRng**
====Definition====
A \emph{function ring} (or $f$\emph{-ring}) is a 
[[lattice-ordered rings|lattice-ordered ring]] $\mathbf{F}=\langle F,\vee,\wedge,+,-,0,\cdot\rangle$ such that 


$x\wedge y=0$, $z\ge 0\ \Longrightarrow\ x\cdot z\wedge y=0$, $z\cdot x\wedge y=0$


Remark: 

====Definition====
==Morphisms==
Let $\mathbf{L}$ and $\mathbf{M}$ be $f$-rings. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $f:L\rightarrow M$ that is a
homomorphism: $f(x\vee y)=f(x)\vee f(y)$, $f(x\wedge y)=f(x)\wedge f(y)$, $f(x\cdot y)=f(x)\cdot f(y)$, $f(x+y)=f(x)+f(y)$.
====Examples====


====Basic results====
The variety of $f$-rings is generated by the class of linearly ordered $\ell$-rings. 
This means $f$-rings are subdirect products of linearly ordered $\ell$-rings, i.e. $f$-rings are representable $\ell$-rings (see e.g. [G. Birkhoff, Lattice Theory, 1967]).

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Congruence distributive]]  |yes, see [[lattices]] |
^[[Congruence extension property]]  | |
^[[Congruence n-permutable]]  |yes, $n=2$, see [[groups]] |
^[[Congruence regular]]  |yes, see [[groups]] |
^[[Congruence uniform]]  |yes, see [[groups]] |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |

====Finite members====

Only the one-element $f$-ring.

====Subclasses====
[[Commutative function rings]] 

====Superclasses====
[[Lattice-ordered rings]] 


====References====

[(Ln19xx>
)]

Trace: • dense_linear_orders • multisets • first-order_classes • modules_over_a_ring • vector_spaces • boolean_monoids

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