caa a22 4500 469026839 CHVBK 20180323132732.0 cr unu---uuuuu 170328e19920301xx s 000 0 eng 10.1007/BF01190753 doi (NATIONALLICENCE)springer-10.1007/BF01190753 Sturm Teo Department of Mathematics and Applied Mathematics, University of Natal, King George V Avenue, 4001, Durban, South Africa aut On irreducible elements of semilattices [Elektronische Daten] [Teo Sturm] The concept of (join)-irreducible elements works well, especially for distributive lattices. Therefore our definition of elements of a given degree of irreducibility employs the notion of distributivity as much as possible, even if the irreducibility is defined for elements of a (meet)-semilattice. Via the lattice of hereditary subsets of the poset ofk-irreducible elements of a semilattice (wherek is a cardinal) we obtain a new construction of a D1k-reflection (a sort of free distributive extension) of the semilattice, provided that there are sufficiently manyk-irreducible elements. The last property is satisfied, for example, if the original semilattice is the dual of an algebraic lattice [Dilworth and Crawley, 1960], but this condition is too restrictive for semilattices. It turns out that, under certain limitations, the D1k-reflection of a semilattice both preserves and reflects the degree of irreducibility. Birkhäuser Verlag, 1992 algebra universalis Birkhäuser-Verlag 29/1(1992-03-01), 16-32 0002-5240 29:1<16 1992 29 12 https://doi.org/10.1007/BF01190753 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01190753 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Sturm Teo Department of Mathematics and Applied Mathematics, University of Natal, King George V Avenue, 4001, Durban, South Africa aut NATIONALLICENCE 773 0- algebra universalis Birkhäuser-Verlag 29/1(1992-03-01), 16-32 0002-5240 29:1<16 1992 29 12 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer