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A real [[Vector space|vector space]]  $  A $
that is simultaneously a [[Lattice|lattice]] is called a vector lattice (or [[Riesz space|Riesz space]]) whenever  $  x \leq  y $(
$  \leq  $
is the lattice order) implies  $  x + z \leq  y + z $
for all  $  z \in A $
and  $  ax \leq  ay $
for all positive real numbers  $  a $. 
If  $  A $
is also an [[Algebra|algebra]] and  $  zx \leq  yz $
and  $  xz \leq  yz $
for all  $  z \in A  ^ {+} $, 
the positive cone of  $  A $, 
then  $  A $
is called an  $  l $-
algebra (a lattice-ordered algebra, Riesz algebra).

A Riesz algebra  $  A $
is called an  $  f $-
algebra ( $  f $
for  "function" ) whenever

$$ 
 \inf  ( x,y ) = 0 \Rightarrow  \inf  ( zx,y ) =  \inf  ( xz,y ) = 0 ,  \forall z \in A  ^ {+} .
$$

This notion was introduced by G. Birkhoff and R.S. Pierce in 1956.

An important example of an  $  f $-
algebra is  $  A = C ( X ) $, 
the space of continuous functions (cf. [[Continuous functions, space of|Continuous functions, space of]]) on some [[Topological space|topological space]]  $  X $. 
Other examples are spaces of Baire functions, measurable functions and essentially bounded functions.  $  f $-
Algebras play an important role in operator theory. The second commutant of a commuting subset of bounded Hermitian operators on some Hilbert space is an  $  f $-
algebra. A [[Linear operator|linear operator]]  $  T $
on some vector lattice  $  A $
is called an orthomorphism whenever  $  T $
is the difference of two positive orthomorphisms; a positive orthomorphism  $  S $
on  $  A $
leaves the positive cone of  $  A $
invariant and satisfies  $   \inf  ( Sx,y ) = 0 $
whenever  $   \inf  ( x,y ) = 0 $. 
The space  $  { \mathop{\rm Orth} } ( A ) $
of all orthomorphisms of  $  A $
is an important example of an  $  f $-
algebra in the theory of vector lattices.

A vector lattice  $  A $
is termed Archimedean if  $  0 \leq  nx \leq  y $(
$  n = 1,2, \dots $) 
implies  $  x = 0 $. 
Archimedean  $  f $-
algebras are automatically commutative and associative. An Archimedean  $  f $-
algebra with unit element is semi-prime (i.e., the only nilpotent element is  $  0 $). 
The latter two properties are nice examples of the interplay between order properties and algebraic properties in an  $  f $-
algebra. Many properties of  $  C ( X ) $
are inherited by an  $  f $-
algebra  $  A $
with a unit element (under some additional completeness condition), such as the existence of the square root of a positive element (if  $  x \in A  ^ {+} $, 
then there exists a unique  $  y \in A  ^ {+} $
such that  $  y ^ {( 2 ) } = x $) 
and the existence of an inverse: if  $  e $
is the unit element of  $  A $
and  $  e \leq  x $, 
then  $  x ^ {- 1 } $
exists in  $  A $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Birkhoff,   R.S. Pierce,   "Lattice-ordered rings"  ''An. Acad. Brasil. Ci.'' , '''28'''  (1956)  pp. 41–69</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,   "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR></table>