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Full-Text Articles in Physical Sciences and Mathematics
The Structure Of Locally Integral Involutive Po-Monoids And Semirings, José Gil-Férez, Peter Jipsen, Siddhartha Lodhia
The Structure Of Locally Integral Involutive Po-Monoids And Semirings, José Gil-Férez, Peter Jipsen, Siddhartha Lodhia
Mathematics, Physics, and Computer Science Faculty Articles and Research
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 …
Structure Theorems For Idempotent Residuated Lattices, José Gil-Férez, Peter Jipsen, George Metcalfe
Structure Theorems For Idempotent Residuated Lattices, José Gil-Férez, Peter Jipsen, George Metcalfe
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery’s characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class has …
Weakening Relation Algebras And Fl2-Algebras, Nikolaos Galatos, Peter Jipsen
Weakening Relation Algebras And Fl2-Algebras, Nikolaos Galatos, Peter Jipsen
Mathematics, Physics, and Computer Science Faculty Books and Book Chapters
FL2-algebras are lattice-ordered algebras with two sets of residuated operators. The classes RA of relation algebras and GBI of generalized bunched implication algebras are subvarieties of FL2-algebras. We prove that the congruences of FL2-algebras are determined by the congruence class of the respective identity elements, and we characterize the subsets that correspond to this congruence class. For involutive GBI-algebras the characterization simplifies to a form similar to relation algebras.
For a positive idempotent element p in a relation algebra A, the double division conucleus image p/A/p is an (abstract) weakening relation algebra, …
Distributive Laws In Residuated Binars, Wesley Fussner, Peter Jipsen
Distributive Laws In Residuated Binars, Wesley Fussner, Peter Jipsen
Mathematics, Physics, and Computer Science Faculty Articles and Research
In residuated binars there are six non-obvious distributivity identities of ⋅,/,∖ over ∧,∨. We show that in residuated binars with distributive lattice reducts there are some dependencies among these identities; specifically, there are six pairs of identities that imply another one of these identities, and we provide counterexamples to show that no other dependencies exist among these.