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Norms on semirings. I. (English) Zbl 1243.16052

Here various norms defined on semirings and rings are studied. The paper contains seven sections containing the notions of i) positive and negative cones, ii) additively cancellative semirings, iii) difference rings, iv) parasemifields of fractions, v) additively cancellative parasemifields vi) semifields, vii) norms on semirings.
The authors here throw light on the fact in connection with possible relation between public key cryptography based on some semimodule and semiring notions. In this regard it is mentioned that there are three basic classes of simple semirings such as additively cancellative, additively idempotent and additively nil of index 2.
The authors here indicate the possible connections of additively cancellative simple semirings with discrete logarithms; also indicate that perhaps with cryptography based on primes can be investigated using norms and seminorms on semirings. A few auxiliary results are noted on semiring-valued norms defined on (semi) rings.
Throughout \(Q=Q(+,\bullet,\leq)\) is a non-trivial linearly ordered commutative and associative semiring. \(Q_{ps}\) and \(Q_{ng}\) stand, respectively, for the positive and negative cones of the ordered semiring.
i) The first section contains results as lemmas, regarding the positive and negative cones of it, about possible common element here together with the union of these, some results on \(M_a=\{x\mid x\leq a+x\}\) and \(N_a=\{y\mid a+y\leq y\}\). The rest part discusses ordered commutative semiring \(Q\) with \(0_Q\in Q\) under the assumption \(Q.0=\{w\}\) and the notions dealt here are bi-ideal-simple, bi-absorbing element, zero-multiplication subring, annihilator of \(Q\) with some ring theoretic analogues, and some unusual ones, additively cancellative semirings.
The next section contains an immediate continuation of the preceding one. The third section contains the notion of difference rings. With \(Q\) additively cancellative the difference ring \(R=\{u-v\mid u,v\in Q\}\) is commutative and associative ring possibly without unit element. A relation \(\leq_R\) is so that it appears as a linear ordering compatible with the addition of the ring \(R\) and relates positive and negative cone with some so-called well behaved characters.
To be noted that the positive cone \(R_{ps}\) is semi-subtractive semiring that is linearly ordered by \(\leq_R\). The authors are of opinion that it is not clear whether there exists a linearly ordered additively cancellative parasemifield \(P\) such that some common characteristics are not true.
The next section with the notion of parasemifield of fractions is another natural intuitive attack on the structure. Keeping this in note authors try to obtain an interesting equivalence such that
i) semiring \(Q\) is multiplicatively cancellative and
ii) the order \(\leq\) is multiplicatively cancellative.
With these conditions \(P\) is chosen as the parasemifield of fractions of \(Q\) and the relation \(\leq_P\) is obtained as a linear ordering compatible with the multiplication of the parasemifield. \(Q\) appears as additively cancellative if \(P\) is so. Interestingly enough it is observed that for \(a,b\in Q\), i) \(a\leq b\), ii) \(a\leq_P b\), iii) \(1\leq_Pb/a\) and iv) \(a/b\leq_P1\) are equivalent. Moreover, additively cancellative property of \(Q\) gives a number of basic results some of which are the inheritance of the character of \(Q\) by \(P\). Coincidence of the positive cone with \(Q\) and the negative cone, with empty negative cone. If the positive and negative cone is non-empty then \(xy=x+xy\) for all \(x,y\in Q\) and \(1_Q\in Q\) then \(1_Q=0_Q\).
The next section is additively cancellative parasemifields. i) With \(Q=P\) is a parasemifield and some equivalent conditions are obtained, though very basic in nature. The sixth section contains results on semifields. And the last section contains \(S=S(+,.)\) a non-trivial commutative and associative semiring. If \(\alpha\) is a multiplicative semigroup homomorphism with \(\alpha(xy)=\alpha(x)\alpha(y)\) for all all \(x,y\in S\), then some trivial or easy results appear in this context some of which are as follows:
A) \(\alpha(x+y)\leq\alpha(x)+\alpha(y)\) for all \(x,y\in S\);
B) \(\alpha(x)+\alpha(y)\leq\alpha(x+y)\) for all \(x,y\in S\);
C) \(\alpha(x+y)=\alpha(x)+\alpha(y)\) for all \(x,y\in S\);
D) \(\alpha(x+y)\leq\max\{\alpha(x),\alpha(y)\}\) for all \(x,y\in S\);
E) \(\max\{\alpha(x),\alpha(y)\}\leq\alpha(x+y)\) for all \(x,y\in S\);
F) \(\alpha(x+y)=\max\{\alpha(x),\alpha(y)\}\) for all \(x,y\in S\).
Some of these are equivalent in the manner cited below:
(A) \(\Rightarrow\alpha(x_1+\cdots+x_n)\leq\alpha(x_1)+\cdots+\alpha(x_n)\) for all \(n\geq 1\) and \(x_1,\dots,x_n\in S\);
(A) \(\Rightarrow\alpha(nx)\leq n\alpha(x)\) for all \(x\in S\) and \(n\geq 1\);
(D) \(\Rightarrow\alpha(x_1+\cdots+x_n)\leq\max\{\alpha(x_1),\dots,\alpha(x_n)\}\) for all \(n\geq 1\) and \(x_1,\dots,x_n\in S\);
(E) \(\Rightarrow\max\{\alpha(x_1),\dots,\alpha(x_n)\}\leq\alpha(x_1+\cdots+x_n)\) for all \(n\geq 1\) and \(x_1,\dots,x_n\in S\);
(F) \(\Rightarrow\alpha(x_1+\cdots+x_n)=\max\{\alpha(x_1),\dots,\alpha(x_n)\}\) for all \(n\geq 1\) and \(x_1,\dots,x_n\in S\).
Some other results are easy consequences of what has been stated above.
Finally the last lemma states results on additively cancellative \(Q\)-semi-subtractive and that \(Q_{ps}=O\) with assumption that for every \(a\in Q\{0_Q\}\) there is a positive integer \(m\) such that \(1_Q\leq ma\), and obtain some equivalence on A, B, and D.

MSC:

16Y60 Semirings
12K10 Semifields
06F25 Ordered rings, algebras, modules
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