Fundamental simplification

Operators combine values to produce new values when their operands meet type and value criteria specified by the operator. The operation is performed numerically when operands are tangible and symbolically if otherwise possible.

Operator Symbols	Operator
+ - ⋅ ÷ // | ^	arithmetic
∧ ∨ ⊕	Boolean
∩ ∪ ∖	set algebraic
⊆ ⊤ ⊄ ⊈	set relational
= ≠ < ≤ > ≥	relational

Left Type	Operator	Right Type	Transformation
scalar	arithmetic	scalar	perform addition, subtraction, multiplication, division, quotient, modulus, exponentiation
collection	arithmetic	scalar	distribute operation and scalar across elements of collection
scalar	arithmetic	collection	distribute operation and scalar across elements of tuple
vector	arithmetic	scalar	apply operation and scalar to each component of vector
real	arithmetic	vector	apply operation and scalar to each component of vector
vector	∘	vector	dot-product
composite	∠	composite	promote operands to vectors; find angle between vectors
real	∠	real	create radial from left (magnitude) and right (angle))
real	ɓ	real	evaluate binomial-choose operation aɓb=a !÷(b !⋅(a-b) !)if operands are integer
real	Boolean	real	evaluate boolean and, or, exclusive or
set	set relative	set	evaluate set relation
set or tuple	set relative	set or tuple	promote set to tuple and evaluate multiset relation
set	set algebraic	set	evaluate set operator
set or tuple	set set algebraic	set or tuple	promote set to tuple and evaluate multiset operation
any type	relational	any type	promote operand with lower type to type of operand with higher type and perform sequence comparison
set	relational	set	perform equality or inequality comparison
real	relational	real	perform numeric comparison
real	radical	real	root(a, b) evaluates root(a, b) if operands are integer and result is integer
variable	→	tuple	create function reference from variable and tuple
any type	‖	any type	if either side is string, coerce operands to string and concatenate
	‖		if either side is matrix, coerce operands to matrix and concatenate
	‖		coerce operands to tuple and concatenate
any type	‗	any type	coerce operands to matrix and concatenate vertically
matrix	×	matrix	perform matrix multiplication
matrix	×	column vector	coerce right operand to matrix, perform matrix multiplication and coerce result to column vector
row vector	×	matrix	coerce left operand to matrix, perform matrix multiplication and coerce result to row vector
row vector	×	row vector	perform 3-dimensional vector cross product and return a row vector
col vector	×	col vector	perform 3-dimensional vector cross product and return a column vector
col vector	×	row vector	perform 3-dimensional vector cross product and return a column matrix
row vector	×	col vector	perform matrix produce and return a the singleton-matrix result as a scalar
string	↓	real	remove leading characters
matrix	↓	real	remove leading rows
matrix	↓	tuple	if tuple has one real element, remove leading columns
matrix	↓	tuple	if tuple has two real elements, remove leading rows and columns
matrix	↓	tuple	if tuple has one set-of-real element, remove rows indexed by elements of set
matrix	↓	tuple	if tuple has two set-of-real elements, remove rows and columns indexed by elements of set
tuple	↓	real	remove leading elements
vector	↓	real	remove leading components
string	↑	real	keep leading characters
matrix	↑	real	keep leading rows
matrix	↑	tuple	if tuple has one real element, keep leading columns
matrix	↑	tuple	if tuple has two real elements, keep leading rows and columns
tuple	↑	real	keep leading elements
vector	↑	real	keep leading components
any type	∈	set	test for set containment
any type	∉	set	test for set exclusion
matrix	⊗	matrix	Kroenecker delta: every element of left matrix multiplied by a copy of the right matrix