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In the definition of a structure (monoid, group, field a.s.o.) some author use the closure property, other authors do not.

For instance Beachy & Blair, Abstract Algebra, do not use the closure in the definition of group (3.1.1), but they use it in the definition of field (4.1.1).

Serge Lang, Algebra, do not use it, neither for monoid, nor for group, nor for ring.

To prove that something is a group or a ring or something else, we have to prove that the defined operations are closed. Thus the closure should be part of the definition.

I think that most authors consider that every binary operation is defined for each pair of elements.

Why should the closure be included in a definition? Or why not?

PeptideChain
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In Lang's book, he defines the (monoid, group, etc) operation on a set $S$ by a map $S\times S \to S$. Closure is implicit because the operation, by the definition of this map, only takes values in $S$.

Quinn Greicius
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    And what about Beachy & Blair? In the definition of field they specifically write binary operations. And previously they say that "A binary operation * on a set S is a rule that assigns to each ordered pair (a , b) of elements of S a unique element a * b of S ." Why have they added a closure axiom in the definition of field? It is the same situation as in Lang, but they made another choice. – PeptideChain Apr 10 '16 at 20:23
  • Are you asking why an explicit mention of closure is made for a field but not for the binary operation on $S$? – Quinn Greicius Apr 10 '16 at 20:27
  • No, I'm asking why B&B in the definition of field have both: 1) the term binary operation (previously defined as closed); and 2) a closure axiom. It is redundant. But perhaps they had their reasons. – PeptideChain Apr 10 '16 at 20:36
  • It is like here: http://math.stackexchange.com/questions/838291/. I do not know where the definition comes from (which author), but there are both elements: the binary operation and the closure axiom (in this case of a group). – PeptideChain Apr 10 '16 at 20:41
  • Under Beachy & Blair's definition of binary operation, the mention of closure is redundant. Moreover, if you look on page 120 of the 3rd edition, there is an earlier definition of a field which doesn't mention closure explicitly. It's probably just there as a reminder of that important fact, even if it doesn't make for the most efficient definition. – Quinn Greicius Apr 10 '16 at 20:53