572 research outputs found
Deciding absorption
We characterize absorption in finite idempotent algebras by means of
J\'onsson absorption and cube term blockers. As an application we show that it
is decidable whether a given subset is an absorbing subuniverse of an algebra
given by the tables of its basic operations
Variations on Algebra: monadicity and generalisations of equational theories
Dedicated to Rod Burstal
Monads with arities and their associated theories
After a review of the concept of "monad with arities" we show that the
category of algebras for such a monad has a canonical dense generator. This is
used to extend the correspondence between finitary monads on sets and Lawvere's
algebraic theories to a general correspondence between monads and theories for
a given category with arities. As application we determine arities for the free
groupoid monad on involutive graphs and recover the symmetric simplicial nerve
characterisation of groupoids.Comment: New introduction; Section 1 shortened and redispatched with Section
2; Subsections on symmetric operads (3.14) and symmetric simplicial sets
(4.17) added; Bibliography complete
Enriched Lawvere Theories for Operational Semantics
Enriched Lawvere theories are a generalization of Lawvere theories that allow
us to describe the operational semantics of formal systems. For example, a
graph enriched Lawvere theory describes structures that have a graph of
operations of each arity, where the vertices are operations and the edges are
rewrites between operations. Enriched theories can be used to equip systems
with operational semantics, and maps between enriching categories can serve to
translate between different forms of operational and denotational semantics.
The Grothendieck construction lets us study all models of all enriched theories
in all contexts in a single category. We illustrate these ideas with the
SKI-combinator calculus, a variable-free version of the lambda calculus.Comment: In Proceedings ACT 2019, arXiv:2009.0633
The Universal Theory of First Order Algebras and Various Reducts
First order formulas in a relational signature can be considered as
operations on the relations of an underlying set, giving rise to multisorted
algebras we call first order algebras. We present universal axioms so that an
algebra satisfies the axioms iff it embeds into a first order algebra.
Importantly, our argument is modular and also works for, e.g., the positive
existential algebras (where we restrict attention to the positive existential
formulas) and the quantifier-free algebras. We also explain the relationship to
theories, and indicate how to add in function symbols.Comment: 30 page
Generalizations of Swierczkowski's lemma and the arity gap of finite functions
Swierczkowski's Lemma - as it is usually formulated - asserts that if f is an
at least quaternary operation on a finite set A and every operation obtained
from f by identifying a pair of variables is a projection, then f is a
semiprojection. We generalize this lemma in various ways. First, it is extended
to B-valued functions on A instead of operations on A and to essentially at
most unary functions instead of projections. Then we characterize the arity gap
of functions of small arities in terms of quasi-arity, which in turn provides a
further generalization of Swierczkowski's Lemma. Moreover, we explicitly
classify all pseudo-Boolean functions according to their arity gap. Finally, we
present a general characterization of the arity gaps of B-valued functions on
arbitrary finite sets A.Comment: 11 pages, proofs simplified, contents reorganize
Weak Cat-Operads
An operad (this paper deals with non-symmetric operads)may be conceived as a
partial algebra with a family of insertion operations, Gerstenhaber's circle-i
products, which satisfy two kinds of associativity, one of them involving
commutativity. A Cat-operad is an operad enriched over the category Cat of
small categories, as a 2-category with small hom-categories is a category
enriched over Cat. The notion of weak Cat-operad is to the notion of Cat-operad
what the notion of bicategory is to the notion of 2-category. The equations of
operads like associativity of insertions are replaced by isomorphisms in a
category. The goal of this paper is to formulate conditions concerning these
isomorphisms that ensure coherence, in the sense that all diagrams of canonical
arrows commute. This is the sense in which the notions of monoidal category and
bicategory are coherent. The coherence proof in the paper is much simplified by
indexing the insertion operations in a context-independent way, and not in the
usual manner. This proof, which is in the style of term rewriting, involves an
argument with normal forms that generalizes what is established with the
completeness proof for the standard presentation of symmetric groups. This
generalization may be of an independent interest, and related to matters other
than those studied in this paper. Some of the coherence conditions for weak
Cat-operads lead to the hemiassociahedron, which is a polyhedron related to,
but different from, the three-dimensional associahedron and permutohedron.Comment: 38 pages, version prepared for publication in Logical Methods in
Computer Science, the authors' last version is v
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