Many-sorted and single-sorted algebras

Mućka, Anna; Romanowska, Anna; Smith, Jonathan D. H.

doi:10.1007/s00012-013-0224-5

Cited by 6 publications

(4 citation statements)

References 18 publications

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“…A precursors of P appears in [2] (see also the last section of [11]). Recently, the functor P was applied by Mučka, Romanowska, and Smith [19] to find good sets of explicit defining identities and quasi-identities for the variety of single-sorted algebras equivalent to the class of all multisorted algebras of a given language where either all sorts are nonempty or all sorts are empty. Freese and McMenzie [9] used a variant of the construction provided by the functor C to associate modules (over appropriate rings) to abelian congruences of algebras in a congruence modular variety, which has played a prominent role in commutator theory for over three decades.…”

Section: Introductionmentioning

confidence: 99%

Algebras from Congruences

Mayr,

Szendrei

2019

Preprint

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We present a functorial construction which, starting from a congruence α of finite index in an algebra A, yields a new algebra C with the following properties: the congruence lattice of C is isomorphic to the interval of congruences between 0 and α on A, this isomorphism preserves higher commutators and TCT types, and C inherits all idempotent Maltsev conditions from A.As applications of this construction, we first show that supernilpotence is decidable for congruences of finite algebras in varieties that omit type 1. Secondly, we prove that the subpower membership problem for finite algebras with a cube term can be effectively reduced to membership questions in subdirect products of subdirectly irreducible algebras with central monoliths. As a consequence, we obtain a polynomial time algorithm for the subpower membership problem for finite algebras with a cube term in which the monolith of every subdirectly irreducible section has a supernilpotent centralizer.

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Section: Introductionmentioning

confidence: 99%

Algebras from Congruences

Mayr,

Szendrei

2019

Preprint

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“…One can cite for instance, application areas such as multisorted algebras (also called many sorted algebras) [5], hidden algebra [6,7], ordered sorted algebra [8]. But other algebraic concepts have also been introduced with quite a different description of their structures as the coalgebras [9].…”

Section: Introductionmentioning

confidence: 99%

Multisorted Tree Algebra

Zobo¹,

Ndjodo²

2014

ACM

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This paper introduces basic concepts describing a hierarchical algebraic structure called multisorted tree algebra. This structure is constructed by placing multisorted algebra at the bottom of a hierarchy and placing at other intermediate nodes the aggregation of algebras placed at their immediate subordinate nodes. These constructions are different from the one of subalgebras, homomorphic images and product algebras used to characterize varieties in universal algebra theory. The resulting hierarchical algebraic structures cannot be easily classified in common universal algebra varieties. The aggregation method and the fundamental properties of the aggregated algebras have been presented with an illustrative example. Multisorted tree algebras spans multisorted algebra concepts and can be used as modelling framework for building hierarchical abstract data types for information processing in organizations.

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“…Recent paper [7] proves that a category of many-sorted algebras on pure sets (Definition 2.8 below) is categorically equivalent to a category of single-sorted algebras. In this paper, we consider on the case of varieties.…”

Section: Introductionmentioning

confidence: 99%

“…Definition 2.9 (cf. [7,Definition 3.1]). The following single-sorted <κ-ary clone C DS is called the clone corresponds to the variety of diagonal algebras of degree S:…”

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confidence: 99%