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We show that every locally integral involutive partially ordered monoid (ipo-monoid) A = (A,⩽, ·, 1,∼,−), and in particular every locally integral involutive semiring, decomposes in a unique way into a family {Ap : p ∈ A+} of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are semirings. Moreover, we show that there is a family of monoid homomorphisms Φ = {φpq : Ap → Aq : p ⩽ q}, indexed on the positive cone (A+,⩽), so that the structure of A can be recovered as a glueing R ΦAp of its integral components along Φ. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family of integral ipo-monoids {Ap : p ∈ D}, indexed on a lower-bounded join-semilattice (D,∨, 1), along a family of monoid homomorphisms Φ is an ipo-monoid.


This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Relational and Algebraic Methods in Computer Science. RAMiCS 2023. Lecture Notes in Computer Science, volume 13896, in 2023. The final publication may differ and is available at Springer via


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