Our main result states that a finite semiring of order > 2 with zero which is not a ring is congruence-simple if and only if it is isomorphic to a ‘dense’ subsemiring of the endomorphism semiring of a finite idempotent commutative monoid. We also investigate those subsemirings further, addressing e.g. the question of isomorphy.

Lower, upper, sandwich, mixed, and convex power domains are isomorphic to domains of second order predicates mapping predicates on the ground domain to logical values in a semiring. The various power domains diier in the nature of the underlying semiring logic and in logical constraints on the second order predicates.

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...