Relation Algebras, Idempotent Semirings and Generalized Bunched Implication Algebras

Jipsen, Peter

doi:10.1007/978-3-319-57418-9_9

Cited by 17 publications

(15 citation statements)

References 15 publications

(16 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our initial motivations for studying idempotents in Q ∨ (I n ) originates from the algebra of logic, see e.g. [14]. Willing to investigate congruences of Q ∨ (I n ) as a residuated lattice [10], it can be shown, using idempotents, that every subalgebra of a residuated lattice Q ∨ (I n ) is simple.…”

Section: Discussionmentioning

confidence: 99%

On Discrete Idempotent Paths

Santocanale

2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

The set of discrete lattice paths from (0, 0) to (n, n) with North and East steps (i.e. words w ∈ { x, y } * such that |w| x = |w| y = n) has a canonical monoid structure inherited from the bijection with the set of join-continuous maps from the chain { 0, 1, . . . , n } to itself. We explicitly describe this monoid structure and, relying on a general characterization of idempotent join-continuous maps from a complete lattice to itself, we characterize idempotent paths as upper zigzag paths. We argue that these paths are counted by the odd Fibonacci numbers. Our method yields a geometric/combinatorial proof of counting results, due to Howie and to Laradji and Umar, for idempotents in monoids of monotone endomaps on finite chains.

show abstract

Section: Discussionmentioning

confidence: 99%

On Discrete Idempotent Paths

Santocanale

2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…The following proposition is also easy to prove from the observation that the join-endomorphisms over a linear order are also monotonic functions. In fact, this result appears in [ 1 ] and it is well-known among the RAMICS community [ 10 , 16 ].…”

Section: The Size Of the Function Spacementioning

confidence: 80%

“…We present proofs for the statements of this section. Propositions 3 and 4 follow from simple observations and they are part of the lattice theory folklore [1,11,16]. We present original proofs of these proposition in the Appendix.…”

Section: Proofsmentioning

confidence: 99%

“…To the best of our knowledge we are the first to establish the cardinality (n + 1) 2 + n!L n (−1) for the lattice M n . The cardinalities n log 2 n for power sets (boolean algebras) and 2n n for linear orders can also found in the lattice literature [1,11,16]. We presented our original proofs of these statements.…”

Section: Conclusion and Related Workmentioning

confidence: 99%

“…Propositions 3 and 4 state, respectively, the size of for the cases when is a powerset lattice and when is a total order. These propositions follow from simple observations and they are part of the lattice theory folklore [ 1 , 10 , 16 ]. We include our original proofs of these propositions in the technical report of this paper [ 13 ].…”

Section: The Size Of the Function Spacementioning

confidence: 99%

See 2 more Smart Citations

Counting and Computing Join-Endomorphisms in Lattices

Quintero

Ramírez

Rueda

et al. 2020

Relational and Algebraic Methods in Computer Science

View full text Add to dashboard Cite

Structures involving a lattice and join-endomorphisms on it are ubiquitous in computer science. We study the cardinality of the set E(L) of all joinendomorphisms of a given finite lattice L. In particular, we show that when L is Mn, the discrete order of n elements extended with top and bottom, |E(L)| = n!Ln(−1) + (n + 1) 2 where Ln(x) is the Laguerre polynomial of degree n. We also study the following problem: Given a lattice L of size n and a set S ⊆ E(L) of size m, find the greatest lower bound E(L) S. The join-endomorphism E(L) S has meaningful interpretations in epistemic logic, distributed systems, and Aumann structures. We show that this problem can be solved with worst-case time complexity in O(n + m log n) for powerset lattices, O(mn 2) for lattices of sets, and O(mn + n 3) for arbitrary lattices. The complexity is expressed in terms of the basic binary lattice operations performed by the algorithm.

show abstract

The Involutive Quantaloid of Completely Distributive Lattices

Santocanale¹

2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Let L be a complete lattice and let be the unital quantale of join-continuous endo-functions of L . We prove that has at most two cyclic elements, and that if it has a non-trivial cyclic element, then L is completely distributive and is involutive (that is, non-commutative cyclic -autonomous). If this is the case, then the dual tensor operation corresponds, via Raney’s transforms, to composition in the (dual) quantale of meet-continuous endo-functions of L . Let be the category of sup-lattices and join-continuous functions and let be the full subcategory of whose objects are the completely distributive lattices. We argue that is itself an involutive quantaloid, thus it is the largest full-subcategory of with this property. Since is closed under the monoidal operations of , we also argue that if is involutive, then is completely distributive as well; consequently, any lattice embedding into an involutive quantale of the form has, as its domain, a distributive lattice.

show abstract

Relation Algebras, Idempotent Semirings and Generalized Bunched Implication Algebras

Cited by 17 publications

References 15 publications

On Discrete Idempotent Paths

On Discrete Idempotent Paths

Counting and Computing Join-Endomorphisms in Lattices

The Involutive Quantaloid of Completely Distributive Lattices

Contact Info

Product

Resources

About