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Boolean matrices ... neither Boolean nor matrices

100%
Ricci G.
Discussiones Mathematicae - General Algebra and Applications
|
2000
|
tom 20
|
nr 1
141-151
EN
Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we "show" that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A minor modification of this standard definition might fix it.
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A semantic construction of two-ary integers

88%
Ricci G.
Discussiones Mathematicae - General Algebra and Applications
|
2005
|
tom 25
|
nr 2
165-219
EN
To binary trees, two-ary integers are what usual integers are to natural numbers, seen as unary trees. We can represent two-ary integers as binary trees too, yet with leaves labelled by binary words and with a structural restriction. In a sense, they are simpler than the binary trees, they relativize. Hence, contrary to the extensions known from Arithmetic and Algebra, this integer extension does not make the starting objects more complex. We use a semantic construction to get this extension. This method differs from the algebraic ones, mainly because it is able to find equational features of the extended objects. Two-ary integers turn out to form the free algebra corresponding to the Jónsson-Tarski's "paradoxical" equations. This entails that they have a "sum" operation as well as other operations of higher dimensions. Two-ary integers can provide LISP memories with convenient direct access jumps and the above low complexity hints at feasible hardware implementations.
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