Koc, Notes on

commutativity of prime rings with generalized derivations, Commun.

Q] are identical modulo associativity and

commutativity of "[conjunction]" and "[disjunction]".

Implicitly, also, specific properties such as

commutativity are checked.

This note defines a scalar measure of the

commutativity of two orthogonal projections.

Intuitively, the idempotence of + is needed to get a partial order over the elements of the semiring (otherwise we would not have reflexivity); the

commutativity of x allows us to consider sets of constraints (instead of ordered tuples); and the fact that 1 is the absorbing element of + makes the element 1 the maximum element of the partial order.

In the case of L(H), this is equivalent to the

commutativity of the respective projection operators onto the subspaces p and q, and is associated in the physical interpretation with the notion of "simultaneous observability".

Also,

commutativity was assumed; that is, the fare from Toronto to Montreal and return is the same as the airfare from Montreal to Toronto and return.

Additional Key Words and Phrases: Atomic operations,

commutativity analysis, synchronization, optimistic synchronization, parallel computing, parallelizing compilers

For instance,

commutativity of operations makes possible the interleaving (without blocking) of transactions on a given object.

x + y = y + x %

Commutativity of + (x + y) + z = x + (y + z) % Associativity of + n(n(n(x) + y) + n (x + y)) = y % Robbins axiom From a set-theoretic perspective, + can be thought of as union and the function n as complement.

where the

commutativity of the upper diagram follows from the fact that [[psi].

In the course of thus acting on the world and reflecting on his actions, he invents the principle of

commutativity.