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Let $(S,\cdot)$ and $(T,\circ)$ be semigroups (or some algebraic structure with an operation), then they are anti-isomorphic if there exists some $\varphi : S \to T$ such that $$ \varphi(xy) = \varphi(y) \circ \varphi(x). $$

Now for what is this notion useful?

The notion of isomorphism is useful as this basically says that isomorphic structures are structurally the same. Anti-Isomorphism in some sense accounts for the order in which we combine elements, so if two structures are isomorphic or anti-isomorphic, what I guess this means is that they are structurally equal without respect to notions which depend on the order of the operation.

In terms of the multiplication table, if we have one set (or a bijection to view both sets as one) and two operations on this set, then both algebraic structures are anti-isomorphic if the transpose (i.e. mirroring it at the diagonal) gets me the other multiplication table.

But all examples that come to my mind are just simple observations, like that the inversion is an anti-isomorphism in every group, or that if we have two finite automata and construct the transformation monoids, we have to choose some order in which we read function composition (from left to right or from right to left), but regardless of what we choose the resulting transformation monoids are anti-isomorphic (but in general not isomorphic).

But despite this mere observations, I do not see where they are really useful; or where this notion is essential?

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  • $\begingroup$ You find it in passing form a finite dimensional vector space to its dual, and $\varphi$ is transposition. $\endgroup$ – Bernard Jun 10 '16 at 19:03
  • $\begingroup$ @Bernhard A vector space and its dual are isomorphic, not anti-isomorphic (but which is the same as addition is commutative)? $\endgroup$ – StefanH Jun 10 '16 at 19:14
  • $\begingroup$ You're right. What I had in mind was transposition is an anti-isomorphism of of the category of finite dimensional vector spaces to itself. $\endgroup$ – Bernard Jun 10 '16 at 19:25
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Well, ... at the very least there are indeed some examples for that! I am going to mention some anti-automorphisms. (Post-) Composing with any isomorphism gives an anti-isomorphism.

  • Taking inverses in a group: $(ab)^{-1} = b^{-1}a^{-1}$
  • Taking the transpose in a semigroup of $n\times n$-matrices: $(AB)^T = B^TA^T$
  • Taking inverses of relations: $(L\circ R)^{-1} = R^{-1}\circ L^{-1}$

There are propably more noteworthy examples, but I would say the importance of contravariance is more obvious, if you indeed study contravariant (semi-) functors in general. For example, one could say it is essential for formalizing the notion of an adjunction, which may be one of the most important concepts in all of mathematics (I dare say). But again more specifically you may be interested in dagger categories (all of the above examples are dagger (semi-) categories with a single object)

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Not sure if this is relevant, but the category-theoretic notion of anti-isomorphism can be helpful when thinking about orderings:

Consider the category $L$ of subsets of size $r$ of $[n]=\{1,2,\ldots,n\}$, where for $A,B\in L, A\to B$ iff $A\leq B$ in lexicographic order. Then there is an anti-isomorphism $\phi$ from $L$ to the category $C$ of the same sets under colexigraphic order defined by $$\phi(A) = \{n+1-a:a\in A\}.$$

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