Normalization of fuzzy \(k\)-ideals in semirings (Q2747302)

Semirings being less restrictive than rings permit a greater variety of ideal-types than rings do, and equally, fuzzy semirings exhibit the same property relative to rings. Thus \(k\)-ideals beget fuzzy \(k\)-ideals and these normal fuzzy \(k\)-ideals \(\mu\), such that \(\mu(x)\) and hence \(\mu(0)\) takes on the crisp value \(1\), where \(\mu(x)\geq\min\{\max\{\mu(x+y),\mu(y+x)\},\mu(y)\}\) for all \(x,y\) in the semiring \(R\) is the defining condition for fuzzy \(k\)-ideals. The normalization condition \(\mu^+(x)=\mu(x)+1-\mu(0)\) is nicely natural and permits certain `obvious' results to be obtained, which as those familiar with obstacles which undermine easy-looking propositions in this area is often much more than meets the eye.