A join-irreducible element is an element in a partially ordered set (poset) that cannot be expressed as the join (supremum) of two other distinct elements. This concept is crucial for understanding the structure of lattices, particularly in distributive and modular lattices, where these elements can signify certain boundaries within the lattice framework. Join-irreducible elements help identify the 'building blocks' of the lattice and can relate to the representation of lattices through certain dualities.
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