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Cancellation property
Cancellation property
Cancellation property
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In mathematics, the notion of cancellativity (or cancellability) is a generalization of the notion of invertibility that does not rely on an inverse element.

An element a in a magma (M, ∗) has the left cancellation property (or is left-cancellative) if for all b and c in M, a ∗ b = a ∗ c always implies that b = c.

An element a in a magma (M, ∗) has the right cancellation property (or is right-cancellative) if for all b and c in M, b ∗ a = c ∗ a always implies that b = c.

An element a in a magma (M, ∗) has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative.

A magma (M, ∗) is left-cancellative if all a in the magma are left cancellative, and similar definitions apply for the right cancellative or two-sided cancellative properties.

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.

Interpretation

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To say that an element in a magma (M, ∗) is left-cancellative, is to say that the function g : x ↦ a ∗ x is injective where x is also an element of M.[1] That the function g is injective implies that given some equality of the form a ∗ x = b, where the only unknown is x, there is only one possible value of x satisfying the equality. More precisely, we are able to define some function f, the inverse of g, such that for all x, f(g(x)) = f(a ∗ x) = x. Put another way, for all x and y in M, if a ∗ x = a ∗ y, then x = y.[2]

Similarly, to say that the element a is right-cancellative, is to say that the function h : x ↦ x ∗ a is injective and that for all x and y in M, if x ∗ a = y ∗ a, then x = y.

Examples of cancellative monoids and semigroups

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The positive (equally non-negative) integers form a cancellative semigroup under addition. The non-negative integers form a cancellative monoid under addition. Each of these is an example of a cancellative magma that is not a quasigroup.

Any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid that embeds into a group (as the above examples clearly do) will obey the cancellative law.

In a different vein, (a subsemigroup of) the multiplicative semigroup of elements of a ring that are not zero divisors (which is just the set of all nonzero elements if the ring in question is a domain, like the integers) has the cancellation property. This remains valid even if the ring in question is noncommutative and/or nonunital.

Non-cancellative algebraic structures

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Although the cancellation property holds for addition and subtraction of integers, real and complex numbers, it does not hold for multiplication due to exception of multiplication by zero. The cancellation property does not hold for any nontrivial structure that has an absorbing element (such as 0).

Whereas the integers and real numbers are not cancellative under multiplication, with the removal of 0, they each form a cancellative structure under multiplication.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Cancellation property

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from Grokipedia
In , the cancellation property is a key structural feature of certain algebraic systems, such as magmas, groups, rings, and fields, that permits the elimination of a common factor from both sides of an equality involving a , provided the factor is nonzero where applicable. For a left cancellation property, if a⋅b=a⋅ca \cdot b = a \cdot c holds in the structure, then b=cb = c; the right cancellation property is defined analogously as b⋅a=c⋅ab \cdot a = c \cdot a implying b=cb = c. This property ensures reliable algebraic manipulations and is closely tied to the absence of zero divisors in multiplicative settings, distinguishing structures like integral domains from more general rings where it may fail. In groups, the cancellation property is universal and holds for every element due to the existence of unique inverses. Specifically, for any g,h1,h2g, h_1, h_2 in a group GG, if gh1=gh2g h_1 = g h_2, then left-multiplying both sides by g−1g^{-1} yields h1=h2h_1 = h_2; the right cancellation follows similarly. This makes groups particularly well-behaved for solving equations and underpins many theorems in group theory, such as applications. In rings, cancellation is equivalent to the ring having no zero divisors: a nonzero element rr allows left cancellation if and only if there is no nonzero ss such that rs=0r s = 0, and likewise for right cancellation, with the two forms being interchangeable in any ring. Commutative rings with unity and no zero divisors—known as integral domains—possess this property fully, enabling simplifications like those in the integers Z\mathbb{Z}. Fields, as special integral domains where every nonzero element has a multiplicative inverse, inherit strong cancellation: if ab=aca b = a c in a field FF, then either a=0a = 0 or b=cb = c, proven by factoring a(b−c)=0a(b - c) = 0 and using invertibility if a≠0a \neq 0. Beyond these core contexts, the cancellation property appears in module theory, where a module AA has it if A⊕B≅A⊕CA \oplus B \cong A \oplus C implies B≅CB \cong C, influencing classifications in . Its failure in non-cancellative structures, like matrix rings over fields, highlights the boundaries of algebraic invertibility and motivates the study of domains and division rings.

Definition and Interpretation

Formal Definition

In , the cancellation property concerns on sets and generalizes the idea of invertibility without requiring inverse elements. Consider a set SS equipped with a ∗:S×S→S*: S \times S \to S, which forms a structure known as a (S,∗)(S, *). The left cancellation property holds in this magma if, for all a,b,c∈Sa, b, c \in S, the equation a∗b=a∗ca * b = a * c implies b=cb = c. Similarly, the right cancellation property holds if b∗a=c∗ab * a = c * a implies b=cb = c for all a,b,c∈Sa, b, c \in S. The two-sided (or simply cancellative) property is satisfied when both left and right cancellation hold simultaneously. These properties can be expressed formally using logical notation. For left cancellation: ∀a,b,c∈S (a∗b=a∗c  ⟹  b=c).\forall a, b, c \in S \ (a * b = a * c \implies b = c). The right cancellation condition is analogous: ∀a,b,c∈S (b∗a=c∗a  ⟹  b=c).\forall a, b, c \in S \ (b * a = c * a \implies b = c). Such definitions assume only the existence of the and do not presuppose additional structure like associativity or identity elements. The concept of the cancellation property emerged in the late , notably in Heinrich Weber's 1882 definition of abstract groups as finite associative systems satisfying left and right cancellation laws.

Variants: Left, Right, and Two-Sided Cancellation

In algebraic structures like magmas, the cancellation property manifests in three distinct variants: left, right, and two-sided. The left cancellation property holds if, for all elements a,b,ca, b, c in the magma (M,⋅)(M, \cdot), a⋅b=a⋅ca \cdot b = a \cdot c implies b=cb = c. The right cancellation property holds if b⋅a=c⋅ab \cdot a = c \cdot a implies b=cb = c for all a,b,c∈Ma, b, c \in M. The two-sided cancellation property is defined as the simultaneous satisfaction of both left and right cancellation. Two-sided cancellation is a stronger condition than either left or right cancellation alone, as it implies both variants, but the converse does not hold in general magmas, where a structure may possess left cancellation without right cancellation, or vice versa. In magmas with commutative operations, where a⋅b=b⋅aa \cdot b = b \cdot a for all a,b∈Ma, b \in M, the left and right variants coincide due to the of the operation, making all three forms equivalent. For instance, under commutativity, a⋅b=a⋅ca \cdot b = a \cdot c rearranges to b⋅a=c⋅ab \cdot a = c \cdot a, so left cancellation directly entails right cancellation. In finite semigroups, left and right cancellation together (constituting two-sided cancellation) imply that the structure is a group, wherein all variants hold uniformly. In monoids, these variants interact with the to enable certain embeddings, though details depend on the specific structure.

Occurrence in Algebraic Structures

In Semigroups and Monoids

In semigroups, which are associative binary operations on a set without requiring an , the cancellation property manifests as left cancellation (for all a,b,ca, b, c, if ab=acab = ac then b=cb = c), right cancellation (if ba=caba = ca then b=cb = c), or two-sided cancellation if both hold. A semigroup is termed cancellative if it satisfies two-sided cancellation. A fundamental result states that every finite cancellative semigroup is in fact a group, as the finiteness combined with cancellation ensures the existence of identities and inverses for all elements. This theorem, originally established by Mal'cev, highlights how cancellation imposes strong structural constraints in finite settings. For infinite semigroups, embeddability into groups requires additional conditions beyond mere cancellativity. Lambek's theorem (1951) characterizes such embeddability: a cancellative can be embedded into a group if and only if it satisfies the Lambek conditions, which are a countable family of identities ensuring the existence of suitable quotients. These conditions complement earlier work and underscore that two-sided cancellation alone is insufficient for embedding, as counterexamples exist where cancellative semigroups fail to embed without further restrictions. In s, which are semigroups equipped with an , cancellation is defined analogously, but the identity facilitates interpretations relative to it, such as distinguishing proper divisors. Free monoids, including the monoid of numbers under (the commutative monoid on one generator), are prototypical examples of cancellative monoids, where the absence of relations preserves cancellation. (1931) extends embeddability results to monoids: a cancellative monoid satisfying the Ore condition—for any a,ba, b, there exist c,dc, d such that ac=bdac = bd—embeds naturally into a group of fractions. For commutative monoids specifically, cancellativity alone suffices for embedding into an , as commutativity implies the Ore condition. In commutative monoids, the cancellation property plays a pivotal role in factorization theory. It implies unique factorization into irreducibles in certain cases, such as when the monoid is free (e.g., Nk\mathbb{N}^k under componentwise , where multisets of generators are unique). Numerical semigroups—finitely generated submonoids of (N,+)(\mathbb{N}, +) with finite complement and gcd of generators equal to 1—provide another context, as they are inherently cancellative and support a rich theory of factorizations into minimal generators, though uniqueness typically requires additional structure like being a .

In Groups, Rings, and Modules

In groups, the presence of inverses ensures that the two-sided cancellation property holds automatically. Specifically, for any elements a,b,ca, b, c in a group GG, if ab=acab = ac, then multiplying both sides on the left by a−1a^{-1} yields b=cb = c. A symmetric argument applies to the right cancellation law: if ba=caba = ca, then right-multiplying both sides by a−1a^{-1} yields b=cb = c. This property distinguishes groups from more general algebraic structures like semigroups, where cancellation may fail without inverses. In rings, cancellation typically refers to the multiplicative structure, particularly in commutative rings with identity. A commutative ring RR with identity is an integral domain if and only if its multiplicative monoid (excluding the zero element) is cancellative, meaning that for all nonzero a,b,c∈Ra, b, c \in R, if ab=acab = ac then b=cb = c, and similarly for right cancellation. This equivalence holds because the absence of zero divisors prevents nontrivial solutions to a(b−c)=0a(b - c) = 0 with a≠0a \neq 0. For example, the ring of integers Z\mathbb{Z} is an integral domain, so it satisfies multiplicative cancellation: if nm=nkn m = n k with n≠0n \neq 0, then m=km = k. In modules, cancellation manifests primarily in the additive structure, as modules over a ring RR form abelian groups under , inheriting the two-sided from groups: for any module elements a,b,ca, b, c, if a+b=a+ca + b = a + c then b=cb = c, and similarly for right cancellation, via (or adding the ). More advanced contexts involve projective modules and exact sequences, where cancellation laws relate to stability under direct sums. A seminal result is Bass's , which states that for a commutative RR of dd and a projective RR-module PP of constant rank greater than dd, if P⊕M≅P⊕NP \oplus M \cong P \oplus N for modules M,NM, N, then M≅NM \cong N. This , established in the , resolves the cancellation problem for high-rank projectives over such rings and has implications for algebraic .

Properties and Implications

Embedding into Groups

One of the most significant implications of the cancellation property in algebraic structures is the potential to embed cancellative s and monoids into groups, where every element has an inverse. This embedding allows the extension of the operation to a fully invertible setting, facilitating deeper analysis through group-theoretic tools. While cancellativity is necessary for such embeddings, it is not sufficient on its own, as demonstrated by Mal'cev's construction of a cancellative that cannot be embedded into any group. In 1939, Mal'cev provided a complete : a embeds into a group it is cancellative and satisfies an infinite system of quasi-identities known as Mal'cev's conditions. These conditions, which involve equations of the form xy=xz  ⟹  yw=zwx y = x z \implies y w = z w generalized over finite lengths, ensure the absence of relations that would prevent inversion in a . This theorem, while precise, is often impractical due to the infinitude of conditions; no finite subset suffices for embeddability. A more constructive and widely used sufficient condition is Ore's condition, introduced in the context of but applicable to . A SS satisfies the right Ore condition if, for all a,b∈Sa, b \in S, the intersection aS∩bS≠∅aS \cap bS \neq \emptyset. For a right cancellative satisfying this condition, guarantees an into a group of right fractions. The group GG is constructed as the set of equivalence classes of pairs (a,b)∈S×S(a, b) \in S \times S, where equivalence is given by (a,b)∼(c,d)(a, b) \sim (c, d) if there exists e∈Se \in S such that ae=cea e = c e and be=deb e = d e. Multiplication is defined using the Ore condition: given (a,b)(a, b) and (c,d)(c, d), there exist x,y∈Sx, y \in S such that bx=ycb x = y c, and (a,b)(c,d)=(ax,yd)(a, b)(c, d) = (a x, y d). The embedding maps s∈Ss \in S to the class of (s,e)(s, e) for a fixed identity ee if SS is a , or adjusted accordingly for ; right cancellativity ensures this map is injective. For commutative cancellative s, the Ore condition holds automatically: given a,b∈Ma, b \in M, take elements such that ab=baa b = b a, placing a common multiple in both principal ideals. Thus, every commutative cancellative monoid embeds into an via the above construction, which coincides with the K(M)K(M). In the Grothendieck construction, assuming additive notation for clarity, K(M)K(M) consists of formal differences −- for m,n∈Mm, n \in M, with relations ensuring −=−- = - if and only if m+q=n+pm + q = n + p, and the embedding m \mapsto - {{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}} is faithful due to cancellativity. A canonical example is the monoid (N,+)(\mathbb{N}, +) of natural numbers under addition, which embeds into the group (Z,+)(\mathbb{Z}, +) of integers. Here, the Grothendieck group identifies positive differences with integers, extending addition invertibly while preserving the original structure. This embedding illustrates how cancellation enables the "universal" group completion, central to applications in algebraic K-theory and beyond.

Relation to Divisibility and Ideals

In commutative monoids, the cancellation property implies that the divisibility relation, defined by a≤ba \leq b if there exists cc such that a⋅c=ba \cdot c = b, is antisymmetric, thereby forming a partial order on the monoid elements. This antisymmetry follows from the fact that if a≤ba \leq b and b≤ab \leq a, then cancellativity ensures a=ba = b, distinguishing cancellative monoids from more general preordered structures where divisibility may only be a preorder. In rings, the cancellative multiplicative structure of the non-zero elements is closely tied to the absence of zero divisors, a defining feature of . Specifically, a with identity is an if and only if its of non-zero elements satisfies the cancellation property: for all non-zero a,b,ca, b, c, if ab=aca b = a c then b=cb = c. This equivalence underscores the foundational role of cancellation in , as explored in classical treatments of . Principal ideal domains (PIDs) exemplify rings where this cancellative structure integrates seamlessly with ideal theory: every ideal is , and the field of fractions inherits the of non-zero elements, in which cancellation holds as a group property. In PIDs, the divisibility relation on non-zero elements aligns with unique , reinforcing the antisymmetry inherited from the underlying structure. Conversely, non-cancellative rings often feature non-trivial ideals that absorb elements, leading to absorption phenomena in products; for instance, matrix rings over fields, such as Mn(F)M_n(\mathbb{F}) for n>1n > 1 and field F\mathbb{F}, lack cancellation due to non-zero matrices A,BA, B with AB=0A B = 0, and while they are simple (with only trivial two-sided ideals), their left and right ideals illustrate absorption in non-commutative settings.

Examples and Counterexamples

Cancellative Examples

In semigroups, the set of positive natural numbers under , denoted N∖{0}\mathbb{N} \setminus \{0\} with the operation ×\times, exemplifies a cancellative . For any a,b,c∈N∖{0}a, b, c \in \mathbb{N} \setminus \{0\}, if a×b=a×ca \times b = a \times c, then b=cb = c, as multiplication by a nonzero integer permits unique division in the positives; the same holds for right cancellation. Similarly, the set of nonnegative integers under addition, (N0,+)(\mathbb{N}_0, +), is a cancellative semigroup, where a+b=a+ca + b = a + c implies b=cb = c due to the injective nature of addition. Another semigroup example is the set of under , (R>0,+)(\mathbb{R}_{>0}, +), which satisfies both left and right cancellation, as addition in reals is invertible and the structure embeds into the additive group of all reals. In monoids, the free monoid on a nonempty AA, consisting of all finite words over AA under with the empty word as identity, is cancellative. Distinct words cannot cancel to the same result, ensuring that if u⋅v=u⋅wu \cdot v = u \cdot w, then v=wv = w, and analogously for right cancellation. All groups exhibit the cancellation property inherently, as the existence of inverses allows solving equations like ab=aca b = a c by left-multiplying by a−1a^{-1}, yielding b=cb = c. For instance, the integers under addition, (Z,+)(\mathbb{Z}, +), form a group where cancellation holds trivially. In rings, fields such as the rational numbers Q\mathbb{Q} with addition and multiplication are integral domains, hence possess the cancellation property: for nonzero a∈Qa \in \mathbb{Q}, ab=aca b = a c implies b=cb = c. Likewise, polynomial rings over integral domains, such as Z\mathbb{Z}, are themselves integral domains, inheriting cancellation; if DD is an integral domain, then DD has no zero divisors, ensuring the property.

Non-Cancellative Examples

In semigroups, the set of all 2×2 matrices over the real numbers under matrix multiplication forms a non-cancellative example. Non-invertible matrices lead to failures of both left and right cancellation; for instance, consider the matrices
A=(1000),B=(1111),C=(1100).A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}.
Then AB=AC=(1100)A B = A C = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}, but B≠CB \neq C, violating left cancellation. Similarly, right cancellation fails due to the presence of zero divisors in the semigroup structure. The of all functions from a with at least two elements to itself, under , is another non-cancellative structure. Constant functions are non-injective and cause cancellation to fail; specifically, if ff is a and g≠hg \neq h are distinct functions such that f∘g=f∘hf \circ g = f \circ h (which occurs when the images of gg and hh are contained in the singleton set that ff maps to), then composition does not allow cancellation. This illustrates how non-injective elements disrupt the property in transformation monoids. In rings, the ring Z/6Z\mathbb{Z}/6\mathbb{Z} under provides a clear of cancellation due to zero divisors. For elements a=2+6Za = 2 + 6\mathbb{Z}, b=3+6Zb = 3 + 6\mathbb{Z}, and c=0+6Zc = 0 + 6\mathbb{Z}, we have a⋅b=0=a⋅ca \cdot b = 0 = a \cdot c, but b≠cb \neq c, so left multiplication by aa (a nonzero element) does not cancel. This example highlights how composite moduli introduce zero divisors that prevent the cancellation property from holding in the multiplicative of the ring. For modules over domains (PIDs), cancellation in direct sums—M ⊕ N ≅ M ⊕ P implying N ≅ P—fails in the non-finitely generated case. Over Z\mathbb{Z} (a PID), consider M = ℤ, N = ℤ ⊕ ℤ, and P = ⨁{n=1}^∞ ℤ (the infinite direct sum of copies of ℤ). Then M ⊕ P ≅ ℤ ⊕ ⨁{n=1}^∞ ℤ ≅ ⨁{n=0}^∞ ℤ ≅ (ℤ ⊕ ℤ) ⊕ ⨁{n=2}^∞ ℤ ≅ N ⊕ P, but M ≇ N since one has and the other has rank 2. Free modules of infinite rank, particularly those involving infinite direct sums, thus exhibit this breakdown, with implications for classifying modules beyond the finitely generated setting. In non-commutative geometry, certain non-commutative algebras, such as deformed Weyl algebras or quantum coordinate rings, can fail cancellation properties analogous to those in classical settings, leading to non-unique decompositions in their module categories; for example, counterexamples arise in the study of projective modules over non-commutative affine spaces where stable does not imply . These structures underscore the challenges in extending classical cancellation to quantum geometries.

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