A heterobasidium is a basidium of the Heterobasidiomycetes. A homobasidium is a basidium of the Homobasidiomycetes. Source: Ainsworth and Bisby's Dictionary of the Fungi, 1983
The basidium of the homobasidiomycetes is a usually more or less clavate undivided cell (= without septae), whereas the basidium of a heterobasidiomycete is a cell divided longitudinal or transverse in (usually) four parts by three septae
Heterobasidiomycetes mainly refers to fungi that produce secondary spore by repetition, for example Tulasnella spp. Sebacina spp. Atractiellales and others. However, now is not very common to use this nomenclature.
No. Heterobasidiomycetes refer to basidiomycetes that produce basidia that are divided longitudinally or transversely (as Andreas Gminder stated above). The longitudinal form occurs in the jelly fungi and appears cruciate when viewed from above the basidium. The transverse form is typical of a germinated ustilospore of the rusts.
I do not know what is meant by "secondary spore by repetition" unless this is referring to anamorphic spores, in which case, it has nothing to do with the distinction between hetero- and homo- basidiomycetes.
basidia with transverse septae also occure in the Auriculariales, e. g. the "chinese morels" = Mu-Err = Auricularia polytricha agg. and also in Helicobasidium.
I think what Dario means is the secondary spore production by spores acting like probasidia, in getting a sterigmum and producing basidiospores.
May be Dario refers to the definition of WELLS (1994) were heterobasidiomycetes CAN have sepatate basidia, and if not, then at least repetative spores. Homobasidiomycetes are consiquently defined as having homogenuous basidia AND non-repetative spores. This circumscription is not unanimousely accepted. Others see the genera with non-septate basidia but repetative spores in the homobasidiomycetes. E. g. Tulasnella belongs in the Cantharellales, and also Ceratobasidium is not seen as a heterobasidiomycete by most mycologists.
B.t.w., WELLS separates the rust and smut from all other fungi (also the heterobasidiomycets) by the simple construction of the dolipore..
Dear colleagues, In the next text taken from Webster and Weaber 2007, you can fin one deifinition that include more than one character to define this concept.
"The class Heterobasidiomycetes is approximately synonymous with the terms ‘Phragmobasidio- mycetes’ or ‘jelly fungi’ and contains fungi with the following characteristics.
1. The dolipore septum is complex, i.e. it is surrounded by a parenthesome. Parenthesomes are also found in the Homobasidiomycetes (Chap- ters 19 and 20), but not in the Urediniomycetes (Chapter 22) and Ustilaginomycetes (Chapter 23).
2. The basidia of Heterobasidiomycetes may be strongly lobed and often divided by transverse, oblique or longitudinal septa. Such basidia are loosely termed heterobasidia, especially if they arise directly from hyphae instead of teliospores as in most Urediniomycetes and Ustilaginomy- cetes. If the basidia are septate, they are also called phragmobasidia. The sterigma of the heterobasidium is unusually prominent and is often termed epibasidium (Martin, 1945). In contrast, the basidia of Homobasidiomycetes are club-shaped and always single-celled.
3. The fruit bodies of most Heterobasidio- mycetes are simpler in architecture than those of Homobasidiomycetes, and the hymenium is not normally protected by a roof- or shelf-like archi- tecture. In compensation, these simple fruit bodies are generally able to survive drying and rehydration, with fresh crops of basidiospores produced after each rehydration event. Fully hydrated basidiocarps are typically greatly swollen and gelatinous, hence the term ‘jelly fungi’ for the Heterobasidiomycetes.
4. The basidiospores of most species are capable of producing secondary spores which may be ballistoconidia, passively released conidia or yeast cells."
The latter part is about the production of secondary basidiospores.
I don't intended to submit the feeling, that you (@Dario) had made mistakes, but wanted to stress that there are different views of the definition of heterobasidiomycetes.
The characters given by Webster & Weaber are good all in all, but (like always in fungi) you will find rare examples which don't fit the concept and mark transitions to other classes (or taxonomical units). Xerotolerant species occure in other groups as well, but it is correct that in basidiomycetes its nearly only the jelly fungi which have this capability. But not all, e. g. Pseudohydnum, Tremellodendropsis and Helicobasidium are not xerotolerant.
But it's always easy to find one exception to the rule. But were to put the frontiers from one taxonomic unit to another one (here heterobasidiomyctees to homobasidiomycetes), that is often a matter of weighting different characters - and you can have arguments for this as well as for another weighting, without than one of them is "a mistake".
We have a proverb here saying "Our knowledge we have today is a mistake seen tomorrow" :-)
Many regrets for contradicting your opinions, but there are neither Homobasidiomycetes, nor Homobasidiomycetes in up-to-date Basidiomycota taxonomy. There are 3 subdivisions (according to Hibbett at al., 2007 and many others) which comprise classes with homobasidia along with heterobasidia intermixed (i.e. Agaricomycetes). Heterobasidia differ from homobasidia in its ontogeny implying 2 steps from initial cell vs 1. Practically heterobasidia are differ in its shape from homobasidia being Y-shaped (unicelled, Dacrymycetes) or with transverse or longitudinal septae (Tremellomycetes, Agaricomycetes: Auriculariales, Cantharellales)
There are two pairs of terms, confused in many cases.
These oppositions are: 1) homobasidium-heterobasidium and 2) holobasidium-phragmobasidium.
1. Heterobasidium is a basidium with prominent hypobasidial and epibasidial parts. Hypobasidial part is a remnant of probasidium where karyogamy took place. Epibasidial part is so-called metabasidium, the sprout where postmeiotic nuclei are expanded through mitoses. Heterobasidium can be unseptate (i.e. holobasidium) or septate (phragmobasidium). Septation appeared on metabsidial stage. An examples of unseptated heterobasidia are dacrymycetoid forked basidia, tulasnelloid basidia bearing 4 epibasidia.
2. Homobasidium is a basidium where hypobasidial and epibasidial parts are weakly differentiated. The reason is elimination of probasidium stage - zygote is divided without any dormant period. All homobasidia are holobasidia, i.e. are devoid of septation. Only adventive septa can be occured within collapsed homobasidia.