Finally, both [PIM.sup.right arrow.sub.t] and [PIM.sup.right arrow.sub.t] produce normal forms modulo associativity and commutativity
of [conjunction] and [disjunction], i.e., with respect to (B8), (B9), (B12), and (B13).
is clear and the finite generation follows from Hilbert's Basis Theorem.
Since the commutativity
of transformations depends on their semantic properties, such commutation fails only if there is a true order dependency on transformations.
Note that: Dislike the Neutrosophic crisp dilation operator, the Neutrosophic crisp erosion does not satisfy commutativity
and the associativity properties.
For a 2-simplex [phi] as above, the existence of 3-simplices with the boundary appropriate for [s.sub.0]([phi]) and [s.sub.1]([phi]) amounts to the commutativity
of the two diagrams in the definition of 2-simplices in [M.sub.1].
A Compensatory Fuzzy Logic system is a quartet of operators: a conjunction, a disjunction, a negation and a strict fuzzy order that satisfies the axioms of compensation, commutativity
, strict growth, veto, fuzzy reciprocity, fuzzy transitivity and the Morgan's law .
39) noted that "Arrays provide a vehicle for teachers to focus students' attention on the nature of the quantities involved, the associated language, the relationship between multiplication and division, and commutativity
Also note that the last equation corresponds to the commutativity
of the meridian and longitude for the component [l.sub.2], and so, can be obtained from the other relations.
The proposed security measure evaluates the similarity between two vague sets of cover images and stego images in terms of n-order Markov chain for capturing the interpixel correlation and has been shown to have the properties of boundedness, commutativity
, and unity.
With the change of variables x = s/(s + i) and y = s + t and using the commutativity
, we get
It is easy to see that the unbalanced linguistic generalized arithmetic Heronian mean operator does not satisfy the property of commutativity