Homomorphic Image and Inverse Image of Weak Closure Operations on Ideals of BCK-Algebras

We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras φ : X → Y , we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping c l Y : I ( Y ) → I ( Y ) , we define a map c l Y ← on I ( X ) by A ↦ φ − 1 ( φ ( A ) c l Y ) . We prove that, if “ c l Y ” is a weak closure operation (respectively, semi-prime and meet) on I ( Y ) , then so is “ c l Y ← ” on I ( X ) . In addition, for mapping c l X : I ( X ) → I ( X ) , we define a map c l X → on I ( Y ) as follows: B ↦ φ ( φ − 1 ( B ) c l X ) . We show that, if “ c l X ” is a weak closure operation (respectively, semi-prime and meet) on I ( X ) , then so is “ c l X → ” on I ( Y ) .

Primeness of Relative Annihilators in BCK-Algebra

Conditions that are necessary for the relative annihilator in lower B C K -semilattices to be a prime ideal are discussed. Given the minimal prime decomposition of an ideal A, a condition for any prime ideal to be one of the minimal prime factors of A is provided. Homomorphic image and pre-image of the minimal prime decomposition of an ideal are considered. Using a semi-prime closure operation “ c l ”, we show that every minimal prime factor of a c l -closed ideal A is also c l -closed.

Disjunctive Multiple-Conclusion Consequence Relations

The concept of multiple-conclusion consequence relation from [8] and [7] is considered. The closure operation C assigning to any binary relation r (dened on the power set of a set of all formulas of a given language) the least multiple-conclusion consequence relation containing r, is dened on the grounds of a natural Galois connection. It is shown that the very closure C is an isomorphism from the power set algebra of a simple binary relation to the Boolean algebra of all multiple-conclusion consequence relations.

Ǝ-closure and Alternatives

This remark considers the interaction of Alternative Semantics (AS) with various binding operations—centrally, Predicate Abstraction (PA) and Ǝ-closure; less centrally, intensionalization. Contra Griffiths’s (2019) theory of ellipsis, I argue that it is technically problematic to appeal to the inherent incompatibility of PA and AS, while assuming the compatibility of Ǝ-closure and AS. I show that the formal pressures that characterize the interaction of PA and alternatives apply equally to Ǝ-closure and alternatives, and that it is accordingly impossible to define a true Ǝ-closure operation within what might be termed “standard” AS. A well-behaved AS reflex of Ǝ-closure can only be defined in compositional settings where a well-behaved AS reflex of PA is definable too. I consider various technical and empirical consequences of these points for Griffiths’s theory of ellipsis, and for linguistic theory more generally.

Cases of low anterior resection and ileostomy for rectal cancer that required more than 18 months for stoma closure

Laparoscopic surgery is performed worldwide, even for cases of rectal cancer close to the anus, and advances in surgical instruments and techniques have increased the number of cases for which anastomosis can be performed, even those cases for which abdominoperineal resection was performed previously. Consequently, as a measure to avoid complications in the event of suture failure after surgery, the number of cases of establishing diverting stoma has also increased. Diverting ostomy may require a closure operation earlier than planned due to colostomy complications, cases requiring a long period of time until closure due to postoperative complications, and cases in which closure operation cannot be performed. Herein, we report cases that took more than 36 months to allow closure of the diverting stoma.

A case study in programming coinductive proofs: Howe’s method

Bisimulation proofs play a central role in programming languages in establishing rich properties such as contextual equivalence. They are also challenging to mechanize, since they require a combination of inductive and coinductive reasoning on open terms. In this paper, we describe mechanizing the property that similarity in the call-by-name lambda calculus is a pre-congruence using Howe’s method in the Beluga formal reasoning system. The development relies on three key ingredients: (1) we give a higher order abstract syntax (HOAS) encoding of lambda terms together with their operational semantics as intrinsically typed terms, thereby avoiding not only the need to deal with binders, renaming and substitutions, but keeping all typing invariants implicit; (2) we take advantage of Beluga’s support for representing open terms using built-in contexts and simultaneous substitutions: this allows us to directly state central definitions such as open simulation without resorting to the usual inductive closure operation and to encode very elegantly notoriously painful proofs such as the substitutivity of the Howe relation; (3) we exploit the possibility of reasoning by coinduction in Beluga’s reasoning logic. The end result is succinct and elegant, thanks to the high-level abstractions and primitives Beluga provides. We believe that this mechanization is a significant example that illustrates Beluga’s strength at mechanizing challenging (co)inductive proofs using HOAS encodings.

A Syntactic Approach to Closure Operation

In the paper, tracing the traditional Hilbert-style syntactic account of logics, a syntactic characteristic of a closure operation defined on a complete lattice follows. The approach is based on observation that the role of rule of inference for a given consequence operation may be played by an ordinary binary relation on the complete lattice on which the closure operation is defined.