Module lattices::algebra

Module for definiting algebraic structures and properties.

Functions

• abelian_group
  Defines an abelian group structure. An abelian group is a group where the operation f is commutative.
• absorbing_element
  Defines the absorbing_element property. An element z is absorbing if az = z and za = z for all a.
• associativity
  Defines the associativity property. a(bc) = (ab)c
• bilinearity
  Defines the bilinearity property q is bilinear with respect to + if q(a + b, c) = q(a,c) + q(b,c) and q(a,c + d) = q(a,c) + q(a,d) This is the same as q being distributive over the addition operation of the three groups S, T, and R in q:S x T –> R As defined in the paper “DBSP: Automatic Incremental View Maintenance for Rich Query Languages Input parameters f, h, g, and q represent (f) the base operation of the algebraic structure on the left input to the query q, (h) the base operation of the algebraic structure on the right input to the query q, (g) the base operation of the algebraic structure the query q outputs to, and (q) The query over (f,g) that we want to check for bilinearity (to incrementalize)
• commutative_monoid
  Defines a commutative monoid structure. A commutative monoid is a monoid where the operation f is commutative.
• commutative_ring
  Defines a commutative ring structure. A commutative ring is a ring where the multiplication operation g is commutative.
• commutativity
  Defines the commutativity property. xy = yx
• distributive
  Defines the distributive property a(b+c) = ab + ac and (b+c)a = ba + ca
• field
  Defines a field structure. A field is a commutative ring where every element has a multiplicative inverse.
• get_single_function_properties
  Loop through each algebraic property in SINGLE_FUNCTION_PROPERTIES and test for them
• group
  Defines a group structure. A group is a set of items along with an associative binary operation f an identity element zero and every element has an inverse element with respect to f
• idempotency
  Defines the idempotency property. xx = x
• identity
  Defines the identity property. An element e is the identity of f if ae = a and ea = a for all a.
• integral_domain
  Defines an integral domain structure. An integral domain is a nonzero commutative ring with no nonzero zero divisors.
• inverse
  Defines the inverse property. An element b is the inverse of a if ab = e and ba = e for some identity element e.
• left_distributes
  Defines the left distributive property a(b+c) = ab + ac
• linearity
  Defines the linearity property q is linear with respect to some group operation + if q(a+b) = q(a) + q(b) This is the same as q being a group homomorphism As defined in the paper “DBSP: Automatic Incremental View Maintenance for Rich Query Languages” Input parameters f, g, and q represent (f) the base operation of the algebraic structure for state, (g) the base operation of the algebraic structure the query q outputs to and (q) the query over f that we want to check for linearity (to incrementalize) respectively
• monoid
  Defines a monoid structure. A monoid is a set of items along with an associative binary operation f and an identity element zero. The f operation combines two items and the zero element acts as the identity for f.
• no_nonzero_zero_divisors
  Defines a no-nonzero-zero-divisors property. x is a nonzero divisor if xy = 0 and y is also a nonzero element.
• nonzero_inverse
  Defines the non_zero inverse property. Every element except zero must have an inverse.
• right_distributes
  Defines the right distributive property. (b+c)a = ba + ca
• ring
  Defines a ring structure. A ring is a semiring with an inverse operation on f.
• semigroup
  Defines a semigroup structure. A semigroup is a set of items along with an associative binary operation f.
• semiring
  Defines a semiring structure. A semiring is a set of items along with two associative binary operations f and g, and two identity elements zero and one. f must be commutative and g must distribute over f. the zero of f must also be absorbing with respect to g.