(maxima.info)Introduction to operators

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7.1 Introduction to operators
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It is possible to define new operators with specified precedence, to
undefine existing operators, or to redefine the precedence of existing
operators.  An operator may be unary prefix or unary postfix, binary
infix, n-ary infix, matchfix, or nofix.  "Matchfix" means a pair of
symbols which enclose their argument or arguments, and "nofix" means an
operator which takes no arguments.  As examples of the different types
of operators, there are the following.

unary prefix
     negation '- a'
unary postfix
     factorial 'a!'
binary infix
     exponentiation 'a^b'
n-ary infix
     addition 'a + b'
matchfix
     list construction '[a, b]'

   (There are no built-in nofix operators; for an example of such an
operator, see 'nofix'.)

   The mechanism to define a new operator is straightforward.  It is
only necessary to declare a function as an operator; the operator
function might or might not be defined.

   An example of user-defined operators is the following.  Note that the
explicit function call '"dd" (a)' is equivalent to 'dd a', likewise
'"<-" (a, b)' is equivalent to 'a <- b'.  Note also that the functions
'"dd"' and '"<-"' are undefined in this example.

     (%i1) prefix ("dd");
     (%o1)                          dd
     (%i2) dd a;
     (%o2)                         dd a
     (%i3) "dd" (a);
     (%o3)                         dd a
     (%i4) infix ("<-");
     (%o4)                          <-
     (%i5) a <- dd b;
     (%o5)                      a <- dd b
     (%i6) "<-" (a, "dd" (b));
     (%o6)                      a <- dd b

   The Maxima functions which define new operators are summarized in
this table, stating the default left and right binding powers (lbp and
rbp, respectively).  (Binding power determines operator precedence.
However, since left and right binding powers can differ, binding power
is somewhat more complicated than precedence.)  Some of the operation
definition functions take additional arguments; see the function
descriptions for details.

'prefix'
     rbp=180
'postfix'
     lbp=180
'infix'
     lbp=180, rbp=180
'nary'
     lbp=180, rbp=180
'matchfix'
     (binding power not applicable)
'nofix'
     (binding power not applicable)

   For comparison, here are some built-in operators and their left and
right binding powers.

     Operator   lbp     rbp

       :        180     20
       ::       180     20
       :=       180     20
       ::=      180     20
       !        160
       !!       160
       ^        140     139
       .        130     129
       *        120
       /        120     120
       +        100     100
       -        100     134
       =        80      80
       #        80      80
       >        80      80
       >=       80      80
       <        80      80
       <=       80      80
       not              70
       and      65
       or       60
       ,        10
       $        -1
       ;        -1

   'remove' and 'kill' remove operator properties from an atom.  'remove
("<a>", op)' removes only the operator properties of <a>.  'kill
("<a>")' removes all properties of <a>, including the operator
properties.  Note that the name of the operator must be enclosed in
quotation marks.

     (%i1) infix ("##");
     (%o1)                          ##
     (%i2) "##" (a, b) := a^b;
                                          b
     (%o2)                     a ## b := a
     (%i3) 5 ## 3;
     (%o3)                          125
     (%i4) remove ("##", op);
     (%o4)                         done
     (%i5) 5 ## 3;
     Incorrect syntax: # is not a prefix operator
     5 ##
       ^
     (%i5) "##" (5, 3);
     (%o5)                          125
     (%i6) infix ("##");
     (%o6)                          ##
     (%i7) 5 ## 3;
     (%o7)                          125
     (%i8) kill ("##");
     (%o8)                         done
     (%i9) 5 ## 3;
     Incorrect syntax: # is not a prefix operator
     5 ##
       ^
     (%i9) "##" (5, 3);
     (%o9)                       ##(5, 3)