GENERALIZED DERIVED ALGEBRAS AND GENERALIZED INDUCED ALGEBRAS

Substituting for the fundamental operations of an algebra term operations we get a new algebra of the same type, called a generalized derived algebra. Such substitutions are called generalized hypersubstitutions. Generalized hypersubstitutions can also be applied to every equation of a fully invariant equational theory. The equational theory generated by the resulting set of the equations induces on every algebra of the type under consideration a fully invariant congruence relation. If we factorize the generalized derived algebra by this fully invariant congruence relation we will obtain an algebra which we call generalized induced algebra. In this paper, we prove some properties which transfer the starting algebras to generalized derived algebras and to generalized induced algebras.

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On equivalence of semigroup identities

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-14321 ◽

2001 ◽

Vol 88 (2) ◽

pp. 161 ◽

Cited By ~ 1

Author(s):

O. Macedońska ◽

M. Żabka

Keyword(s):

Free Semigroup ◽

Positive Answer ◽

The Other ◽

Invariant Congruence ◽

Semigroup Identities ◽

Fully Invariant Congruence

For a given relation $\rho$ on a free semigroup ${\mathcal F}$ we describe the smallest cancellative fully invariant congruence ${\rho}^{\sharp}$ containing $\rho$. Two semigroup identities are s-equivalent if each of them is a consequence of the other on cancellative semigroups. If two semigroup identities are equivalent on groups, it is not known if they are s-equivalent. We give a positive answer to this question for all binary semigroup identities of the degree less or equal to 5. A poset of corresponding varieties of groups is given.

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EQUATIONAL THEORIES GENERATED BY HYPERSUBSTITUTIONS OF TYPE (n)

International Journal of Algebra and Computation ◽

10.1142/s0218196702001218 ◽

2002 ◽

Vol 12 (06) ◽

pp. 867-876 ◽

Cited By ~ 2

Author(s):

K. DENECKE ◽

J. KOPPITZ ◽

ST. NIWCZYK

Keyword(s):

Free Term ◽

Operation Symbol ◽

Arbitrary Type ◽

Equational Theories ◽

Invariant Congruence ◽

Congruence Relations ◽

Term Algebra ◽

Fully Invariant Congruence

Hypersubstitutions map n-ary operation symbols to n-ary terms. Such mappings can be uniquely extended to mappings defined on the set of all terms. It turns out that the kernels of hypersubstitutions are fully invariant congruence relations on the (absolutely free) term algebra of the considered type. For an arbitrary type τ = (n), n ≥ 1, i.e. if one has only one n-ary operation symbol, we will describe all these congruence relations. The results will be applied to solve the hyperunification problem. Further we will give some generalizations to arbitrary types.

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Congruence relations on almost distributive lattices-II

Asian-European Journal of Mathematics ◽

10.1142/s1793557121500054 ◽

2019 ◽

pp. 2150005

Author(s):

Gezahagne Mulat Addis

Keyword(s):

Distributive Lattice ◽

Congruence Relation ◽

Congruence Class ◽

Distributive Lattices ◽

Ideal Formula ◽

Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

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POSITIVE FRAGMENTS OF RELEVANCE LOGIC AND ALGEBRAS OF BINARY RELATIONS

The Review of Symbolic Logic ◽

10.1017/s1755020310000249 ◽

2010 ◽

Vol 4 (1) ◽

pp. 81-105 ◽

Cited By ~ 4

Author(s):

ROBIN HIRSCH ◽

SZABOLCS MIKULÁS

Keyword(s):

Equational Theory ◽

Relevance Logic ◽

Binary Relations ◽

Finitely Based ◽

Similarity Type ◽

Relation Composition

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.

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A note on the equational theory of modular ortholattices

Algebra Universalis ◽

10.1007/s000120050178 ◽

2000 ◽

Vol 44 (1-2) ◽

pp. 165-168 ◽

Cited By ~ 4

Author(s):

Christian Herrmann ◽

Michael S. Roddy

Keyword(s):

Equational Theory

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States and Measures on Hyper BCK-Algebras

Journal of Applied Mathematics ◽

10.1155/2014/397265 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Xiao-Long Xin ◽

Pu Wang

Keyword(s):

Congruence Relation ◽

Basic Properties ◽

Mv Algebra ◽

Regular Congruence ◽

Bosbach State

We define the notions of Bosbach states and inf-Bosbach states on a bounded hyper BCK-algebra(H,∘,0,e)and derive some basic properties of them. We construct a quotient hyper BCK-algebra via a regular congruence relation. We also define a∘-compatibledregular congruence relationθand aθ-compatibledinf-Bosbach stateson(H,∘,0,e). By inducing an inf-Bosbach states^on the quotient structureH/[0]θ, we show thatH/[0]θis a bounded commutative BCK-algebra which is categorically equivalent to an MV-algebra. In addition, we introduce the notions of hyper measures (states/measure morphisms/state morphisms) on hyper BCK-algebras, and present a relation between hyper state-morphisms and Bosbach states. Then we construct a quotient hyper BCK-algebraH/Ker(m)by a reflexive hyper BCK-idealKer(m). Further, we prove thatH/Ker(m)is a bounded commutative BCK-algebra.

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A call-by-need lambda calculus with locally bottom-avoiding choice: context lemma and correctness of transformations

Mathematical Structures in Computer Science ◽

10.1017/s0960129508006774 ◽

2008 ◽

Vol 18 (3) ◽

pp. 501-553 ◽

Cited By ~ 20

Author(s):

DAVID SABEL ◽

MANFRED SCHMIDT-SCHAUSS

Keyword(s):

Operational Semantics ◽

Lambda Calculus ◽

Equational Theory ◽

Order Reduction ◽

Single Step ◽

Program Transformations ◽

Normal Order ◽

Choice Context ◽

Contextual Equivalence ◽

Fairness Condition

We present a higher-order call-by-need lambda calculus enriched with constructors, case expressions, recursive letrec expressions, a seq operator for sequential evaluation and a non-deterministic operator amb that is locally bottom-avoiding. We use a small-step operational semantics in the form of a single-step rewriting system that defines a (non-deterministic) normal-order reduction. This strategy can be made fair by adding resources for book-keeping. As equational theory, we use contextual equivalence (that is, terms are equal if, when plugged into any program context, their termination behaviour is the same), in which we use a combination of may- and must-convergence, which is appropriate for non-deterministic computations. We show that we can drop the fairness condition for equational reasoning, since the valid equations with respect to normal-order reduction are the same as for fair normal-order reduction. We develop a number of proof tools for proving correctness of program transformations. In particular, we prove a context lemma for both may- and must- convergence that restricts the number of contexts that need to be examined for proving contextual equivalence. Combining this with so-called complete sets of commuting and forking diagrams, we show that all the deterministic reduction rules and some additional transformations preserve contextual equivalence. We also prove a standardisation theorem for fair normal-order reduction. The structure of the ordering ≤c is also analysed, and we show that Ω is not a least element and ≤c already implies contextual equivalence with respect to may-convergence.

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A proof of Zhil'tsov's theorem on decidability of equational theory of epigroups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2155 ◽

2016 ◽

Vol Vol. 17 no. 3 (Combinatorics) ◽

Author(s):

Inna Mikhaylova

Keyword(s):

Finite Semigroup ◽

Equational Theory ◽

Unary Operation ◽

Finitely Based ◽

Infinite Basis ◽

International Audience

International audience Epigroups are semigroups equipped with an additional unary operation called pseudoinversion. Each finite semigroup can be considered as an epigroup. We prove the following theorem announced by Zhil'tsov in 2000: the equational theory of the class of all epigroups coincides with the equational theory of the class of all finite epigroups and is decidable. We show that the theory is not finitely based but provide a transparent infinite basis for it.

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