Graph connectivity theory is important in network implementations, transportation, routing and tolerance, among other things. Separation edges vertices refer to single points of failure a network, so they are often sought-after. Chandramouleeswaran et al. introduced the principle semiring valued graphs, also known as S-valued symmetry 2015. Since then, works on graphs such vertex dominating set...

We present some general results implying nonfinite axiomatisability of many additively idempotent semirings with finitely based semigroup reducts. The smallest is a $3$-element commutative example, which we show also has \texttt{NP}-hard membership for its variety. As well as being the only axiomatisable ai-semiring on $3$-elements, are able to that basis property infects related semirings, inc...

Abstract We study permutability properties of matrix semigroups over commutative bipotent semirings (of which the best-known example is tropical semiring ). prove that every such semigroup weakly permutable (a result previous stated in literature, but with an erroneous proof) and then proceed to depth question when they are strongly (which turns out depend heavily on semiring). Along way we cla...