pylogic.structures.grouplike package¶

Submodules¶

pylogic.structures.grouplike.group module¶

class pylogic.structures.grouplike.group.AbelianGroup[source]¶

class pylogic.structures.grouplike.group.AbelianGroup(*args, **kwargs)

Bases: Group

op_is_commutative: ForallInSet[ForallInSet[Equals]]¶

classmethod property_op_is_commutative(set_: Set, operation: SpecialInfix[Any, Any, Expr, Expr]) → ForallInSet[ForallInSet[Equals]][source]¶

class pylogic.structures.grouplike.group.Group[source]¶

class pylogic.structures.grouplike.group.Group(*args, **kwargs)

Bases: Monoid

have_inverses: ForallInSet[ExistsUniqueInSet[And[Equals, Equals]]]¶

latin_square: ForallInSet[ForallInSet[And[ExistsUniqueInSet[Equals], ExistsUniqueInSet[Equals]]]]¶

classmethod property_have_inverses(set_: Set, operation: SpecialInfix[Any, Any, Expr, Expr], identity: Any) → ForallInSet[ExistsUniqueInSet[And[Equals, Equals]]][source]¶

This is a theorem because it is a consequence of the Latin square property and the identity property.

classmethod property_latin_square(set_: Set, operation: SpecialInfix[Any, Any, Expr, Expr]) → ForallInSet[ForallInSet[And[ExistsUniqueInSet[Equals], ExistsUniqueInSet[Equals]]]][source]¶

The Latin square property states that, for each a and b in Q, there exist unique elements x and y in Q such that both

a * x = b

y * a = b

Then x = a b (left divide) and y = b / a (right divide)

pylogic.structures.grouplike.loop module¶

class pylogic.structures.grouplike.loop.Loop[source]¶

class pylogic.structures.grouplike.loop.Loop(*args, **kwargs)

Bases: Quasigroup

containment_function(x: Any) → bool[source]¶

has_identity: And[IsContainedIn, ForallInSet[And[Equals, Equals]]]¶

classmethod property_has_identity(set_: Set, operation: SpecialInfix[Term, Term, Expr, Expr], identity: Term) → And[IsContainedIn, ForallInSet[And[Equals, Equals]]][source]¶

pylogic.structures.grouplike.magma module¶

class pylogic.structures.grouplike.magma.AttrNames¶

Bases: dict

is_closed_under_op: str¶

op: str¶

operation: str¶

operation_name: str¶

operation_symbol: str¶

class pylogic.structures.grouplike.magma.CustomAttrNamesKwargs[source]¶

Bases: TypedDict

attr_names: AttrNames¶

containment_function: Callable[[...], bool] | None¶

elements: Iterable | None¶

name: str¶

operation: Callable[[Any, Any], Expr] | None¶

operation_name: str | None¶

operation_symbol: str | None¶

class pylogic.structures.grouplike.magma.Magma[source]¶

class pylogic.structures.grouplike.magma.Magma(*args, **kwargs)

Bases: Set

containment_function(x: Any) → bool[source]¶

is_closed_under_op: ForallInSet[ForallInSet[IsContainedIn]]¶

classmethod property_is_closed_under_op(set_: Set, operation: SpecialInfix[Any, Any, Expr, Expr]) → ForallInSet[ForallInSet[IsContainedIn]][source]¶

pylogic.structures.grouplike.monoid module¶

class pylogic.structures.grouplike.monoid.Monoid[source]¶

class pylogic.structures.grouplike.monoid.Monoid(*args, **kwargs)

Bases: Semigroup

containment_function(x: Any) → bool[source]¶

has_identity: And[IsContainedIn, ForallInSet[And[Equals, Equals]]]¶

classmethod property_has_identity(set_: Set, operation: SpecialInfix[Any, Any, Expr, Expr], identity: Any) → And[IsContainedIn, ForallInSet[And[Equals, Equals]]][source]¶

pylogic.structures.grouplike.quasigroup module¶

class pylogic.structures.grouplike.quasigroup.Quasigroup[source]¶

class pylogic.structures.grouplike.quasigroup.Quasigroup(*args, **kwargs)

Bases: Magma

latin_square: ForallInSet[ForallInSet[And[ExistsUniqueInSet[Equals], ExistsUniqueInSet[Equals]]]]¶

classmethod property_latin_square(set_: Set, operation: SpecialInfix[Any, Any, Expr, Expr]) → ForallInSet[ForallInSet[And[ExistsUniqueInSet[Equals], ExistsUniqueInSet[Equals]]]][source]¶

The Latin square property states that, for each a and b in Q, there exist unique elements x and y in Q such that both

a * x = b

y * a = b

Then x = a b (left divide) and y = b / a (right divide)

pylogic.structures.grouplike.semigroup module¶

class pylogic.structures.grouplike.semigroup.Semigroup[source]¶

class pylogic.structures.grouplike.semigroup.Semigroup(*args, **kwargs)

Bases: Magma

op_is_associative: ForallInSet[ForallInSet[ForallInSet[Equals]]]¶

classmethod property_op_is_associative(set_: Set, operation: SpecialInfix[Any, Any, Expr, Expr]) → ForallInSet[ForallInSet[ForallInSet[Equals]]][source]¶

Module contents¶