A Short Introduction to Clones

Kerkhoff, Sebastian; Pöschel, Reinhard; Schneider, Friedrich Martin

doi:10.1016/j.entcs.2014.02.006

Cited by 23 publications

(6 citation statements)

References 25 publications

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“…From the 4 4 n possible G-connectives of arity n, not all combinations are possible (e.g., when there is a G-conditional, there is a G-negation). One can raise a functional completeness problem: which subset is possible (we know they would have to be clones, see Kerkhoff et al, 2014)? More relevant to our current enterprise, what rank is needed to admit exactly a given subset: to obtain all regular connectives, a 4-valued logic is sufficient, what about various proper subsets of all the Gentzen regular connectives?…”

Section: N1mentioning

confidence: 99%

From many-valued consequence to many-valued connectives

Chemla

Égré

2019

Synthese

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Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multiconclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but also on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.

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Section: N1mentioning

confidence: 99%

From many-valued consequence to many-valued connectives

Chemla

Égré

2019

Synthese

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“…The set of all finite arity functions from a set X to itself is called the the full function clone on X; denoted by O X . We do not permit nullary functions in O X following [6,32]. We define the topology of pointwise convergence on O X in a similar way to the full transformation monoid X X where the subbasic open sets are of the form U (a1,...,an),b = {f ∈ O X : (a 1 , .…”

Section: Another Wee Foray Into the Land Of Clonesmentioning

confidence: 99%

Polish topologies on endomorphism monoids of relational structures

Elliott¹,

Jonušas²,

Mitchell³

et al. 2022

Preprint

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In this paper we present general techniques for characterising minimal and maximal semigroup topologies on the endomorphism monoid End(A) of a countable relational structure A. As applications, we show that the endomorphism monoids of several well-known relational structures, including the random graph, the random directed graph, and the random partial order, possess a unique Polish semigroup topology. In every case this unique topology is the subspace topology induced by the usual topology on the Baire space N N . We also show that many of these structures have the property that every homomorphism from their endomorphism monoid to a second countable topological semigroup is continuous; referred to as automatic continuity. Many of the results about endomorphism monoids are extended to clones of polymorphisms on the same structures.

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“…Firstly, there is little hope of obtaining an informative description of the lattice of all De Morgan clones. While the lattice of all clones on a two-element set is countable and has a fairly simple structure first described by Post [25], already the lattice of clones on a threeelement set has the cardinality of the continuum and its structure is widely assumed not to permit a tractable description [22].…”

Section: Introductionmentioning

confidence: 99%

De Morgan clones and four-valued logics

Přenosil¹

2021

Preprint

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We study clones on a four-element set related to the clone DMA of all term functions of the subdirectly irreducible four-element De Morgan algebra DM4. We find generating sets for the clones of all functions preserving the subalgebras of DM4, the automorphisms of DM4, the truth order and the information order on DM4, as well as clones defined by conjunctions of these conditions. We identify the covers of DMA in the lattice of four-valued clones and describe the lattice of clones above DMA which contain the discriminator function. Finally, observing that each clone above DMA defines an expansion of the fourvalued Belnap-Dunn logic, we classify these clones by their metalogical properties, specifically by their position within the Leibniz and Frege hierarchies of abstract algebraic logic.

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A Short Introduction to Clones

Cited by 23 publications

References 25 publications

From many-valued consequence to many-valued connectives

From many-valued consequence to many-valued connectives

Polish topologies on endomorphism monoids of relational structures

De Morgan clones and four-valued logics

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