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#### Keywords

Key Words and pharses: C(S)-action, complete dual frame, C(S)-consistent complete dual frame, Eilenberg–Moore Algebra. Key Words and phrases: Function semilattice, ideal semilattice, semilattice, distributive lattice, Priestley duality, Scott topology, Lawson topology. Key words and phrases: Projection property, Retraction, Binary relation, Cycle-space.#### Institution

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- Department of Mathematics and Applied Mathematics University of Natal King George V Avenue Durban 4001 South Africa e-mail: cjvanalten@yahoo.com, raftery@scifs1.und.ac.za 1 (%)
- Department of Mathematics and Applied Mathematics University of Natal King George V Avenue Durban 4001 South Africa e-mail: raftery@scifs1.und.ac.za 1 (%)
- Department of Mathematics University of South Africa PO Box 392 Pretoria 0003 South Africa e-mail: salbasdo@alpha.unisa.ac.za 1 (%)
- Department of Mathematics, University of Cape Town, Rondebosch 7700, Republic of South Africa, e-mail: peter0@maths.uct.ac.za 1 (%)
- IREMIA, Université de la Réunion 15, avenue René Cassin, BP 7151, F-97715 Saint Denis Messag. cedex 9 France e-mail: delhomme@univ-Reunion.fr 1 (%)

#### Author

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- Delhommé, C. 1 (%)
- Farley, J. D. 1 (%)
- Ouwehand, P. 1 (%)
- Raftery, J. G. 1 (%)
- Rose, H. 1 (%)

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## On the lattice of varieties of residuation algebras

### Algebra Univeralis (1999-09-01) 41: 283-315 , September 01, 1999

No abstract available

## Ideals of Priestley powers of semilattices

### Algebra Univeralis (1999-09-01) 41: 239-254 , September 01, 1999

### Abstract.

Let *X* be a poset and *Y* an ordered space; *X*^{Y}
$(X^Y_\Sigma, X^Y_\Lambda)$
denotes the poset of continuous order-preserving maps from *Y* to *X* with the discrete (respectively, Scott, Lawson) topology. If *S* is a
$\lor$
-semilattice,
$S^\sigma$
its ideal semilattice, and *T* a bounded distributive lattice with Priestley dual space *P(T)*, it is shown that the following isomorphisms hold:
$(S^{P(T)})^\sigma \cong (S^\sigma) ^{P(T^\sigma)}_\Lambda.$
Moreover,
$$(S^\sigma)^{P(T^\sigma)}_\Lambda \cong (S^\sigma) ^{P(T^\sigma)}$ if and only if $(S^\sigma)^{P(T^\sigma)}_\Lambda = (S^\sigma)^{P(T^\sigma)},$
and sufficient conditions and necessary conditions for the isomorphism to hold are obtained (both necessary and sufficient if *S* is a distributive
$\lor$
-semilattice).

## On Eilenberg–Moore algebras induced by chains

### Algebra Univeralis (1999-09-01) 41: 337-359 , September 01, 1999

### Abstract.

Let *S* be a fixed topological space. The contravariant Hom functor given by *C(X)* =
$Hom_Top(X, S)$
has an adjoint specified, on sets, by *P(A)= S*^{A} and the composite,
$M = C \circ P$
, is a Monad on the category of sets. In this paper we characterize the category of Eilenberg–Moore Algebras associated with *M* in the special case where *S* is a linearly ordered space in its specialization order. The characterization is presented in terms of the notion of a dual frame which admits a *C(S)*-action.

## Lattice varieties with non-elementary amalgamation classes

### Algebra Univeralis (1999-09-01) 41: 317-336 , September 01, 1999

### Abstract.

In this paper it is shown that under certain general conditions, if the amalgamation class of a lattice variety contains a member which does not have a 2-congruence, then the amalgamation class is not closed under ultrapowers or direct products. In particular, the amalgamation class is not first order axiomatizable.

## Projection properties and reflexive binary relations

### Algebra Univeralis (1999-09-01) 41: 255-281 , September 01, 1999

### Abstract.

Given a structure
$\frak M$
for which constant mappings are endomorphisms, consider an inter *n* > 1; the structure is *weakly n-projective* if projections are the only retractions of
$\frak {M^n}$
on its diagonal; it is *strongly n-projective* if projections are its only Hamming *n*-operations (i.e. *n*-ary operations regular with respect to each argument) which are identical on the diagonal. We establish general criteria of “projectivity” for reflexive binary relations. We obtain in particular, for each integer *n*, the existence of strongly *n*-projective (symmetric) graphs.