On congruence extension properties for ordered algebras
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On congruence extension properties for ordered algebras Valdis Laan1 · Sohail Nasir2 · Lauri Tart1
Published online: 28 May 2015 © Akadémiai Kiadó, Budapest, Hungary 2015
Abstract We introduce four extension properties (CEP, QEP, SCEP and SQEP) for ordered algebras, similar to the congruence extension property (CEP) and the strong congruence extension property of usual (unordered) algebras. All four properties turn out to have a description in terms of commutative squares or pullback diagrams. We then use these categorical descriptions to prove an ordered analogue of the well-known relation TP = AP + CEP, namely that a variety of ordered algebras has the ordered transferability property if and only if it has the ordered amalgamation property and QEP. Keywords Ordered algebra · Order-congruence · Compatible quasiorder · Congruence extension property · Quasiorder extension property · Amalgamation property · Transferability property Mathematics Subject Classification
Primary 06F99 · Secondary 18A32 · 08B25
1 Preliminaries As usual in universal algebra, a type is a (possibly empty) set of operation symbols which is a disjoint union of sets k , k ∈ N ∪ {0}.
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Valdis Laan [email protected] Sohail Nasir [email protected] Lauri Tart [email protected]
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Faculty of Mathematics and Computer Science, Institute of Mathematics, University of Tartu, J. Liivi 2, Tartu 50409, Estonia
2
Department of Mathematics, Physics and Geology Cape Breton University, 1250 Grand Lake Rd., Sydney, NS B1P 6L2, Canada
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On congruence extension properties for ordered algebras
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Definition 1.1 (Cf. [1]) Let be a type. An ordered -algebra (or simply ordered algebra) is a triple A = (A, A , ≤ A ) comprising a poset (A, ≤ A ) and a set A of monotone operations on A, where ω A ∈ A has arity k if ω ∈ k and where the monotonicity of ω A means that a1 ≤ A a1 ∧ . . . ∧ ak ≤ A ak ⇒ ω A (a1 , . . . , ak ) ≤ A ω A (a1 , . . . , ak ) for all a1 , . . . , ak , a1 , . . . , ak ∈ A. An ordered algebra B = (B, B , ≤ B ) is called a subalgebra of A = (A, A , ≤ A ) if (i) B ⊆ A, (ii) for every k and for every ω ∈ k , ω B = ω A B k , (iii) ≤ B = ≤ A ∩ (B × B). The order and operations on the direct product of ordered algebras are defined componentwise. A homomorphism f : A −→ B of ordered algebras is a monotone operation-preserving map from an ordered -algebra A to an ordered -algebra B. We call a homomorphism f an order-embedding if additionally f (a) ≤ B f (a ) ⇒ a ≤ A a for all a, a ∈ A. Note that every order-embedding is necessarily injective. An inequality of type is a sequence of symbols t ≤ t with t and t being -terms. , where t , t : An −→ A are the We say that A satisfies inequality t ≤ t if tA ≤ tA A A is defined pointwise. A class K of term functions induced on A by t and t , and tA ≤ tA ordered -algebras is called a variety if it consists precisely of all the algebras satisfying some set of inequalities; we refer to [1] for further details. Every variety of ordered algebras and their homomorphis
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