View source for fa'ai'ai

==== valsi ====
fa'ai'ai
==== type ====
experimental cmavo
==== creator ====
[[personal/krtisfranks|krtisfranks]]
==== time entered ====
Sun Sep  8 05:20:25 2019
 
== English ==
=== Definition #71498 - Preferred ===
 
==== selma'o ====
VUhU
==== definition ====

mekso k-ary operator, for natural k and 1 < k < 5: ordered input  MATH
 $(f, g, S, m)$
 
<span class="MATH">(<i>f</i>, <i>g</i>, <i>S</i>, <i>m</i>)</span> where <span class="MATH"><i>f</i></span> and <span class="MATH"><i>g</i></span> are functions, <span class="MATH"><i>S</i></span> is a set of positive integers or "ro" (="all"), and <span class="MATH"><i>m</i></span> is 0 or 1 (as a toggle); output is a function equivalent to the function <span class="MATH"><i>f</i></span> as applied to an input ordered tuple with <span class="MATH"><i>g</i></span> applied to the entries/terms with indices in <span class="MATH"><i>S</i></span> (or to all entries/terms if <span class="MATH"><i>S</i></span>="ro") if <span class="MATH"><i>m</i> = 0</span>, or <span class="MATH"><i>g</i></span> left-composed with the same if <span class="MATH"><i>m</i> = 1</span>.

==== notes ====

<span class="MATH"><i>S</i></span>="ro" is default case (making this operator binary or ternary); <span class="MATH"><i>S</i></span> must be a set or "ro" (no bare integers); <span class="MATH"><i>m</i></span> = 0 is the default case (making this operator binary or ternary); use "[[mau'au|mau'au]]" and "[[zai'ai|zai'ai]]" in order to quote <span class="MATH"><i>f</i></span> and <span class="MATH"><i>g</i></span> each; the indices in the implicit tuple mentioned in the definition are positive natural numbers such that said tuple is of form  MATH
 $(x_1, x_2, x_3, ..., x_n, ...)$
 
<span class="MATH">(<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>,..., <i>x</i><sub>n</sub>,...)</span>; thus, notationally, a concrete output of this function, as applied to the aforementioned concrete input tuple, is of form <span class="MATH">(</span>fa'ai'ai MATH
 $(f, g, S, m))(x_1, x_2, ..., x_n, ...)$
 
<span class="MATH">(<i>f</i>, <i>g</i>, <i>S</i>, <i>m</i>))(<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>,..., <i>x</i><sub>n</sub>,...)</span>. For example, maintaining this notation, if <span class="MATH"><i>S</i></span>="ro" and <span class="MATH"><i>m</i> = 0</span>, then the output is  MATH
 $f(g(x_1), g(x_2), ..., g(x_n), ...)$
 
<span class="MATH"><i>f</i> (<i>g</i>(<i>x</i><sub>1</sub>), <i>g</i>(<i>x</i><sub>2</sub>),..., <i>g</i>(<i>x</i><sub>n</sub>),...)</span>; if <span class="MATH"><i>S</i></span> = Set(1, 3, 23) and <span class="MATH"><i>m</i> = 0</span>, then the output is  MATH
 $f(g(x_1), x_2, g(x_3), x_4, x_5, ..., x_<a class="undefined" href="../dict/21?bg=1;langidarg=2">21</a>, x_<a class="undefined" href="../dict/22?bg=1;langidarg=2">22</a>, g(x_<a class="undefined" href="../dict/23?bg=1;langidarg=2">23</a>), x_<a class="undefined" href="../dict/24?bg=1;langidarg=2">24</a>, x_<a class="undefined" href="../dict/25?bg=1;langidarg=2">25</a>, ..., x_n, ...)$
 
<span class="MATH"><i>f</i> (<i>g</i>(<i>x</i><sub>1</sub>), <i>x</i><sub>2</sub>, <i>g</i>(<i>x</i><sub>3</sub>), <i>x</i><sub>4</sub>, <i>x</i><sub>5</sub>,..., <i>x</i><sub>21</sub>, <i>x</i><sub>22</sub>, <i>g</i>(<i>x</i><sub>23</sub>), <i>x</i><sub>24</sub>, <i>x</i><sub>25</sub>,..., <i>x</i><sub>n</sub>,...)</span>; vel sim. If <span class="MATH"><i>h</i></span> is the function output by this expression when <span class="MATH"><i>m</i> = 0</span>, then for the same inputs (ignoring <span class="MATH"><i>m</i></span>), <span class="MATH"><i>g</i><tt>o</tt><i>h</i></span> is the function output by the same expression but with <span class="MATH"><i>m</i> = 1</span>. Obviously, in order to be meaningful, the output of each step along the way must be defined. If <span class="MATH"><i>S</i></span> is the empty set and <span class="MATH"><i>g</i></span> is defined, then the output is just the function <span class="MATH"><i>f</i></span> if <span class="MATH"><i>m</i> = 0</span>. The output of this operator is a function, so it must have explicit input supplied to it ("it" here referring to the output of this operator) in order to actually have an explicit and concrete result; use mathematical brackets around this operator and its inputs when doing so, particularly when <span class="MATH"><i>S</i></span> or <span class="MATH"><i>m</i></span> is omitted (as being equal to the default "ro" or 0 resp.).

==== gloss words ====

<ul><li>[[natlang/en/extended functional composition|extended functional composition]]</li></ul>

==== created by ====
[[personal/krtisfranks|krtisfranks]]
==== vote information ====
1
     
==== time ====
Sun Sep  8 06:07:04 2019

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=== Etymology ===