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<head><title>..< -- a binary operator, used for sequences of consecutive items, not including the endpoint</title>
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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="___The_sp__Macaulay2_splanguage.html" title="">The Macaulay2 language</a> > <a href="_operators.html" title="">operators</a> > <a href="_.._lt.html" title="a binary operator, used for sequences of consecutive items, not including the endpoint">..<</a></div>
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<div><h1>..< -- a binary operator, used for sequences of consecutive items, not including the endpoint</h1>
<div class="single"><h2>Description</h2>
<div><p>The most confusing thing about this operator, in all its guises, is that it is not a syntactic construction, and so the resulting sequences do not splice themselves into enclosing lists, as in each of the following examples.</p>
<table class="examples"><tr><td><pre>i1 : {10..<11}
o1 = {1 : (10)}
o1 : List</pre>
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<tr><td><pre>i2 : {10..<8}
o2 = {()}
o2 : List</pre>
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<tr><td><pre>i3 : {3..<5,8..<10}
o3 = {(3, 4), (8, 9)}
o3 : List</pre>
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<p> Use <a href="_splice.html" title="remove subsequences">splice</a> to fix that.</p>
<table class="examples"><tr><td><pre>i4 : splice {3..<5,8..<10}
o4 = {3, 4, 8, 9}
o4 : List</pre>
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<p>If a type of list, instead of a sequence, is desired, use <a href="_to__List.html" title="list of elements">toList</a> or the operator <a href="_new.html" title="new objects and new types">new</a>.</p>
<table class="examples"><tr><td><pre>i5 : 0..<5
o5 = (0, 1, 2, 3, 4)
o5 : Sequence</pre>
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<tr><td><pre>i6 : toList (0..<5)
o6 = {0, 1, 2, 3, 4}
o6 : List</pre>
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<tr><td><pre>i7 : new Array from 0..<5
o7 = [0, 1, 2, 3, 4]
o7 : Array</pre>
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<tr><td><pre>i8 : new Sum from 0..<5
o8 = 0 + 1 + 2 + 3 + 4
o8 : Expression of class Sum</pre>
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<p>The operator can be used with sequences or lists, whose elements are of various types, to produce rectangular intervals.</p>
<table class="examples"><tr><td><pre>i9 : (0,0)..<(2,3)
o9 = ((0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2))
o9 : Sequence</pre>
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<tr><td><pre>i10 : p_(0,a) ..< r_(2,c)
o10 = (p , p , p , p , q , q , q , q )
0,a 0,b 1,a 1,b 0,a 0,b 1,a 1,b
o10 : Sequence</pre>
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<p>Use <a href="_...html" title="a binary operator, used for sequences of consecutive items">..</a> instead to get a sequence that does not stop short of the endpoint.</p>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_ranges_spand_sprepetitions.html" title="">ranges and repetitions</a></span></li>
<li><span><a href="_polynomial_springs.html" title="">polynomial rings</a></span></li>
<li><span><a href="_subscripted_spvariables.html" title="">subscripted variables</a></span></li>
<li><span><a href="_...html" title="a binary operator, used for sequences of consecutive items">..</a> -- a binary operator, used for sequences of consecutive items</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>..<</tt> :</h2>
<ul><li><span><tt>Expression ..< Expression</tt> (missing documentation<!-- tag: (..<,Expression,Expression) -->)</span></li>
<li><span><tt>Expression ..< Holder</tt> (missing documentation<!-- tag: (..<,Expression,Holder) -->)</span></li>
<li><span><tt>Expression ..< Thing</tt> (missing documentation<!-- tag: (..<,Expression,Thing) -->)</span></li>
<li><span><tt>Holder ..< Expression</tt> (missing documentation<!-- tag: (..<,Holder,Expression) -->)</span></li>
<li><span><tt>Holder ..< Holder</tt> (missing documentation<!-- tag: (..<,Holder,Holder) -->)</span></li>
<li><span><a href="___Indexed__Variable_sp.._lt_sp__Indexed__Variable.html" title="sequences of consecutive indexed variables">IndexedVariable ..< IndexedVariable</a> -- sequences of consecutive indexed variables</span></li>
<li><span><tt>IndexedVariableTable ..< IndexedVariableTable</tt> (missing documentation<!-- tag: (..<,IndexedVariableTable,IndexedVariableTable) -->)</span></li>
<li><span><tt>IndexedVariableTable ..< Thing</tt> (missing documentation<!-- tag: (..<,IndexedVariableTable,Thing) -->)</span></li>
<li><span><a href="___List_sp.._lt_sp__List.html" title="rectangular sequences of consecutive lists and sequences">List ..< List</a> -- rectangular sequences of consecutive lists and sequences</span></li>
<li><span>Sequence ..< Sequence, see <span><a href="___List_sp.._lt_sp__List.html" title="rectangular sequences of consecutive lists and sequences">List ..< List</a> -- rectangular sequences of consecutive lists and sequences</span></span></li>
<li><span><tt>MonoidElement ..< MonoidElement</tt> (missing documentation<!-- tag: (..<,MonoidElement,MonoidElement) -->)</span></li>
<li><span><tt>MonoidElement ..< Thing</tt> (missing documentation<!-- tag: (..<,MonoidElement,Thing) -->)</span></li>
<li><span><a href="___Ring__Element_sp.._lt_sp__Ring__Element.html" title="a sequence of consecutive generators of a polynomial ring">RingElement ..< RingElement</a> -- a sequence of consecutive generators of a polynomial ring</span></li>
<li><span><tt>RingElement ..< Thing</tt> (missing documentation<!-- tag: (..<,RingElement,Thing) -->)</span></li>
<li><span><a href="___String_sp.._lt_sp__String.html" title="a sequence of consecutive strings, not including the endpoint">String ..< String</a> -- a sequence of consecutive strings, not including the endpoint</span></li>
<li><span><tt>Subscript ..< Subscript</tt> (missing documentation<!-- tag: (..<,Subscript,Subscript) -->)</span></li>
<li><span><tt>Subscript ..< Thing</tt> (missing documentation<!-- tag: (..<,Subscript,Thing) -->)</span></li>
<li><span><a href="___Symbol_sp.._lt_sp__Symbol.html" title="sequences of consecutive symbols">Symbol ..< Symbol</a> -- sequences of consecutive symbols</span></li>
<li><span><tt>Thing ..< Expression</tt> (missing documentation<!-- tag: (..<,Thing,Expression) -->)</span></li>
<li><span><tt>Thing ..< IndexedVariableTable</tt> (missing documentation<!-- tag: (..<,Thing,IndexedVariableTable) -->)</span></li>
<li><span><tt>Thing ..< MonoidElement</tt> (missing documentation<!-- tag: (..<,Thing,MonoidElement) -->)</span></li>
<li><span><tt>Thing ..< RingElement</tt> (missing documentation<!-- tag: (..<,Thing,RingElement) -->)</span></li>
<li><span><tt>Thing ..< Subscript</tt> (missing documentation<!-- tag: (..<,Thing,Subscript) -->)</span></li>
<li><span><a href="___Z__Z_sp.._lt_sp__Z__Z.html" title="sequences of consecutive integers">ZZ ..< ZZ</a> -- sequences of consecutive integers</span></li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="_.._lt.html" title="a binary operator, used for sequences of consecutive items, not including the endpoint">..<</a> is <span>a <a href="___Keyword.html">keyword</a></span>.</p>
<div><div><p>This operator may be used as a binary operator in an expression like <tt>x..<y</tt>. The user may install <a href="_binary_spmethods.html" title="">binary methods</a> for handling such expressions with code such as</p>
<pre> X ..< Y := (x,y) -> ...</pre>
<p>where <tt>X</tt> is the class of <tt>x</tt> and <tt>Y</tt> is the class of <tt>y</tt>.</p>
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