Motivation: "the commutator gives an indication of the extent to which a certain binary operation fails to be commutative" (http://en.wikipedia.org/wiki/Commutator). For example (courtesy of wikipedia), in a group, $G$, for each $g,h \in G$, the commutator of $g$ and $h$, $[g,h]$ is defined by, $$[g,h] = g^{-1}h^{-1}gh$$

"the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure" (http://en.wikipedia.org/wiki/Associator). For example (thanks wikipedia!), for a nonassociative ring or algebra $R$, the associator is the multilinear map $[\cdot,\cdot,\cdot] : R \times R \times R \to R$ given by, $$[x,y,z] = (xy)z-x(yz)$$

Comment: I was expecting to find an analog to the associator and commutator for the property of distributivity. I've failed to find such an analog on wikipedia and via google searches. This lack of information gives me a sad face.

Question: My question is, does an established analog to the (concept of the) associator and commutator exist for the property of distributivity? If so are there any good sources of information on it and its use?

In certain situations we can define an operator that measures the degree to which one binary operation fails to be (left/right) distributive over another and call it the (left/right) distributor. For example suppose $(S,+,\times)$ is an algebraic structure such that $(S,+)$ is an abelian group and $(S,\times)$ is a semigroup. Then we can define the left distributor, $\mathrm{Ldis} : S \times S \times S \to S$, by $$\mathrm{Ldis}(x,y,z) = xy+xz - x(y+z)$$ Similarly we can define the right distributor, $\mathrm{Rdis} : S \times S \times S \to S$, by $$\mathrm{Rdis}(x,y,z) = yx+zx-(y+z)x$$ where, as per usual, $-$ is used to denote additive inverse, juxtaposition is used in place of $\times$, and $\times$ has higher precedence than $+$. In the case that $\times$ is commutative, $\mathrm{Ldis} = \mathrm{Rdis}$, and we can define the distributor, $\mathrm{dis}$, by setting it equal to the left or right distributor.

Example: In any ring we have, $$\mathrm{dis}(x,y,z) = 0$$

Example: In any wheel algebra we have, $$\mathrm{dis}(x,y,z) = 0x$$

Side question: Can anyone think of any other structures with a distributor-like operator satisfying an identity which does not force said structure to be trivial?

Any help or information on the subject is very much appreciated. Thank you for your time and consideration. I hope you have a great day.



  • $\begingroup$ It's a fine idea, but as your side question hints I don't any examples in which it's useful. $\endgroup$ – Kevin Carlson Apr 7 '15 at 3:24
  • 1
    $\begingroup$ That may be true Kevin. On the other hand, you don't know if something is going to be useful or not all the way up until the moment you find a use for it. :) $\endgroup$ – DAS Apr 7 '15 at 4:14
  • $\begingroup$ Trivial remark: $\mathrm{Ldis}$ can be used to "make" a structure left distributive. If $(R,+,\cdot)$ is a structure with two binary operations, then we can mod out the smallest congruence relation satisfying $x \cdot y + x \cdot z \sim x \cdot (y+z)$. This results in the universal structure of the same type which receives a homomorphism from $(R,+,\cdot)$ and is left distributive. If we want our $(R,+)$ to be an abelian group, and $(R,\cdot)$ be a semigroup, we might as well mod out the smallest ideal containing all $\mathrm{Ldis}(x,y,z)$. $\endgroup$ – Martin Brandenburg Apr 7 '15 at 13:56

The notion of distributor is not new and has been used in near-ring theory for a long time. It was apparently introduced by Fröhlich [2, 3] in 1958. Here are a few other relevant references.

[1] L. Esch, Commutator and Distributor Theory in Near-Rings, Doc. Diss. (Boston Univ., 1974).

[2] A. Fröhlich, Distributively Generated Near-Rings: (I. Ideal Theory) Proc. London Math. Soc. (1958) s3-8 (1): 76-94 doi:10.1112/plms/s3-8.1.76

[3] A. Fröhlich, Distributively Generated Near-Rings: (II. Representation Theory), Proc. London Math. Soc. (1958) s3-8 (1): 95-108 doi:10.1112/plms/s3-8.1.95

[4] J. G. Gutiérrez, Distributor ideals in near-rings of polynomials, Arch. Math. 55 (1990), no. 6, 537-541.

[5] J. D. P. Meldrum, Near-rings and their links with groups. Research Notes in Mathematics, 134. Pitman (Advanced Publishing Program), Boston, MA, 1985. x+275 pp. ISBN: 0-273-08701-0

[6] J.D.P. Meldrum, Les généralisations de la distributivité dans les presque-anneaux. Rend. Sem. Mat. e Fis. Milano LIX (1989), 9-24

[7] G. Pilz, Near-rings, the theory and its applications, Mathematics Studies, No. 23, North-Holland, Amsterdam, 1977, xiv + 393 pp.

  • $\begingroup$ An excellent answer! $\endgroup$ – DAS May 12 '15 at 13:32
  • $\begingroup$ You're welcome. $\endgroup$ – J.-E. Pin May 12 '15 at 13:34
  • $\begingroup$ Hmm... Near-rings all keep either the left or right distributive law intact. Is there a type of structure in which neither the left or right distributive law hold for every triple of elements? $\endgroup$ – DAS Feb 10 '16 at 19:28
  • $\begingroup$ Yes, non distributive lattices, for instance modular lattices. $\endgroup$ – J.-E. Pin Feb 10 '16 at 21:35

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