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In a semigroup, a left-invertible element is left-cancellative, and analogously for right and two-sided. If a −1 is the left inverse of a, then a ∗ b = a ∗ c implies a −1 ∗ (a ∗ b) = a −1 ∗ (a ∗ c), which implies b = c by associativity. For example, every quasigroup, and thus every group, is cancellative.
In mathematics, the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers.. Given an operation denoted here ∗, and an identity element denoted e, if x ∗ y = e, one says that x is a left inverse of y, and that y is a right inverse of x.
A right inverse in mathematics may refer to: A right inverse element with respect to a binary operation on a set; A right inverse function for a mapping between sets;
A loop has the weak inverse property when (xy)z = e if and only if x(yz) = e. This may be stated in terms of inverses via (xy) λ x = y λ or equivalently x(yx) ρ = y ρ. A loop has the inverse property if it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties.
Both use of left/right inverse and section/retraction are commonly seen in the literature: the former use has the advantage that it is familiar from the theory of semigroups and monoids; the latter is considered less confusing by some because one does not have to think about 'which way around' composition goes, an issue that has become greater ...
A magma is a set M matched with an operation • that sends any two elements a, b ∈ M to another element, a • b ∈ M. The symbol • is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation (M, •) must satisfy the following requirement (known as the magma or closure property):
Igneous rock (igneous from Latin igneus 'fiery'), or magmatic rock, is one of the three main rock types, the others being sedimentary and metamorphic. Igneous rocks are formed through the cooling and solidification of magma or lava. The magma can be derived from partial melts of existing rocks in either a planet's mantle or crust.
Magma (from Ancient Greek μάγμα (mágma) 'thick unguent') [1] is the molten or semi-molten natural material from which all igneous rocks are formed. [2] Magma (sometimes colloquially but incorrectly referred to as lava) is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial ...