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In this paper, definitions of $${\mathcal{K}}$$ automata, $${\mathcal{K}}$$ regular languages, $${\mathcal{K}}$$ regular expressions and $${\mathcal{K}}$$ regular grammars based on lattice-ordered semirings are given. It is shown that $${\mathcal{K}}$$ NFA is equivalent to $${\mathcal{K}}$$ DFA under some finite condition, the Pump Lemma holds if $${\mathcal{K}}$$ is finite, and $${{\mathcal{K}}}\epsilon$$ NFA is equivalent to $${\mathcal{K}}$$ NFA. Further, it is verified that the concatenation of $${\mathcal{K}}$$ regular languages remains a $${\mathcal{K}}$$ regular language. Similar to classical cases and automata theory based on lattice-ordered monoids, it is also found that $${\mathcal{K}}$$ NFA, $${\mathcal{K}}$$ regular expressions and $${\mathcal{K}}$$ regular grammars are equivalent to each other when $${\mathcal{K}}$$ is a complete lattice.